tikz-notes.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "94af752c-b3b3-49b9-9bdf-4b1d1bbcd07f",
   "metadata": {},
   "source": [
    "# TikZ 绘制中学几何题\n",
    "\n",
    "chenshuo@chenshuo.com\n",
    "\n",
    "最新版： https://github.com/chenshuo/typeset/blob/master/tikz-notes.ipynb\n",
    "\n",
    "TikZ 有专门的 `tkz-euclide` 宏包，中文手册：https://github.com/jamesfang8499/tkz-euclide-doc-zh-cn ，例子可参考 https://github.com/jamesfang8499/math2/\n",
    "\n",
    "这里用基础的 TikZ 语法来绘制中学几何习题，满足基本的个人需求。\n",
    "\n",
    "本 Jupyter notebook 用了 https://github.com/mkrphys/ipython-tikzmagic.git 扩展"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f4dee69a-c492-4fe9-952c-c1848b49c2ea",
   "metadata": {},
   "outputs": [],
   "source": [
    "# !pip install git+git://github.com/chenshuo/ipython-tikzmagic.git\n",
    "\n",
    "%load_ext tikzmagic"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e5f95800-f450-49c6-9447-60118e94a4dc",
   "metadata": {},
   "source": [
    "## 基础语法"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a27dc6c4-2c30-4a2e-bb24-cb537be6978a",
   "metadata": {},
   "source": [
    "https://tikz.dev/tikz-coordinates"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b299b5e9-abb8-4621-a664-5d1b48fde4f9",
   "metadata": {},
   "source": [
    "### 直角坐标\n",
    "\n",
    "(x, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cbafbee5-3e30-43a4-85be-c506652c9317",
   "metadata": {},
   "source": [
    "3-4-5 直角三角形 right triangle"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "7689d40e-f9c0-44e8-9d1e-d2104466e6c2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%tikz\n",
    "\n",
    "\\draw (0, 3) -- (0, 0) -- (4, 0) -- cycle;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "81e911d0-71e9-4018-a306-8a63e4b1ce9d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%tikz\n",
    "\n",
    "\\draw (0, 3) node [above] {$A$} -- (0, 0) node [below] {$B$} -- (4, 0) node [below] {$C$}  -- cycle;"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c6671807-8a8f-4bca-8391-ed6c51a36e96",
   "metadata": {},
   "source": [
    "等腰三角形 isosceles triangle"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "2bc005a2-2b79-4ed7-9099-85d16630ea09",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%tikz -l angles,quotes\n",
    "\\draw (0, 0) coordinate (B) -- (1, 2.4) coordinate (A) node [midway] {=} node [above] {(1, 2.4)}\n",
    "  -- (2, 0) coordinate (C) node [midway] {=} -- cycle;\n",
    "\\draw pic [draw,\"$\\alpha$\"] {angle=C--B--A};\n",
    "\\draw pic [draw,\"$\\alpha$\"] {angle=A--C--B};"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8f1aff6f-3cc6-49f1-a0b7-f0d3a363a8a1",
   "metadata": {},
   "source": [
    "### 极坐标\n",
    "\n",
    "(angle:radius)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "896b3258-d6a9-4051-ba81-b822e49f35c5",
   "metadata": {},
   "source": [
    "正三角形 equilateral triangle"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "70531d81-53e0-4fd9-a371-e4e0f66e2449",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%tikz\n",
    "\n",
    "\\draw (0, 0) -- (60:2) node [above] {(60:2)} -- (2, 0) -- cycle;"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c3c4ca31-46cf-453b-b821-3e27483ef720",
   "metadata": {},
   "source": [
    "斜边处于水平位置的直角三角形"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "4eb8284c-1478-4e35-a47e-4e5c31a42982",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%tikz -l angles\n",
    "\n",
    "\\draw (-2, 0) coordinate (A) -- (60:2) coordinate (C) -- (2, 0) coordinate (B) -- cycle;\n",
    "\\draw pic [draw,angle radius=3mm] {right angle=A--C--B};"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a886b73a-26be-4fa4-b249-8c8671084c90",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "34b171a9-2c7e-41d3-b344-7e833fdfb1a4",
   "metadata": {},
   "source": [
    "### 中点及延长线\n",
    "\n",
    "https://tikz.dev/tikz-coordinates#sec-13.5.3\n",
    "\n",
    "AB 中点 `($(A)!0.5!(B)$)`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "f7d19eb4-1e21-4c32-b914-03ecb305e595",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%tikz -l calc\n",
    "\n",
    "\\draw (0, 0) coordinate (A) -- (3, 2) coordinate (B) |- (A) coordinate [midway] (C);\n",
    "\\draw [dashed] (C) -- ($(A)!0.5!(B)$);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "ef3d6331-7f0a-490d-abae-f1e3502a70cd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%tikz -l calc\n",
    "\n",
    "\\draw (0, 0) coordinate(A) node [below] {$A$} -- (4, 0) coordinate (B) node [below] {$B$} -- (3, 2.5) coordinate (C) node [above] {$C$} -- cycle;\n",
    "\\draw ($(A)!1/3!(B)$) coordinate (X) node [below] {$X$}\n",
    "  -- ($(B)!1/2!(C)$) coordinate (Y) node [right] {$Y$}\n",
    "  -- ($(C)!1/4!(A)$) coordinate (Z) node [above] {$Z$} -- cycle;"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "49761f23-1a40-4806-8f09-00b004c007e2",
   "metadata": {},
   "source": [
    "AB 延长线 `($(A)!2.0!(B)$)`\n",
    "\n",
    "$|AN|=4|AM|$, $[\\triangle ABC]=1$, What is $[ABNC]$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "bcea60ea-e34b-4bc2-b7ca-8957703db7c6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%tikz -l calc\n",
    "\n",
    "\\draw (0, 0) coordinate(A) node [below] {$A$} -- (2, 0) coordinate (B) node [below] {$B$} -- (1.6, 1.8) coordinate (C) node [above] {$C$} -- cycle;\n",
    "\\draw ($(B)!0.3!(C)$) coordinate (M) node [above right] {$M$};\n",
    "\\draw (A) -- ($(A)!4!(M)$) coordinate (N) node [above right] {$N$};\n",
    "\\draw (C) -- (N) -- (B);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e79dfc2b-d937-4079-b4a7-0dd93817e31c",
   "metadata": {},
   "source": [
    "AB 反向延长线 `($(A)!-1!(B)$)`"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "14b6a9b5-a4e1-4c6a-bb10-d59127bf66ae",
   "metadata": {},
   "source": [
    "### 距离\n",
    "\n",
    "从 A 到 B 距离 A 点 1cm：`($(A)!1cm!(B)$)`\n",
    "\n",
    "https://tikz.dev/tikz-coordinates#sec-13.5.4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "c9d83af8-5108-47ce-ac22-be22572655e5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%tikz -l calc,angles\n",
    "\n",
    "% 角平分线 angle bisector\n",
    "% https://math.stackexchange.com/questions/2032931/bisectrix-of-a-triangle-in-tikz\n",
    "\n",
    "\\draw (0, 0) coordinate (A) node [below] {$A$} -- (2, 0) coordinate (B) node [below] {$B$};\n",
    "\\draw (A) -- (1.8, 1.5) coordinate (C) node [right] {$C$};\n",
    "\\draw [blue] let\n",
    "    \\p1=($(A)!2cm!(B)$),\n",
    "    \\p2=($(A)!2cm!(C)$),\n",
    "    \\p3=($(\\p1)!0.5!(\\p2)$)\n",
    "  in (A) -- (\\p3) coordinate (D) node [right] {$D$};\n",
    "\\draw pic [draw] {angle=B--A--D};\n",
    "\\draw pic [draw, angle radius=6mm,double] {angle=D--A--C};"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "860b1d99-7e15-4d74-9942-0aa37d3dd8a1",
   "metadata": {},
   "source": [
    "### 偏转角\n",
    "\n",
    "`($(A)!0.5!90:(B)$)` 从 A 点出发，向 B 顺时针偏转 90 度的方向，前进 0.5 倍 AB 距离。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "17d124b9-2288-47b1-936d-2f85289fb7a4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%tikz -l calc,angles\n",
    "\n",
    "% 垂直平分线 perpendicular bisector\n",
    "\n",
    "\\draw (0, 0) coordinate (A) -- (3, 2) coordinate (B);\n",
    "\\fill ($(A)!0.5!(B)$) coordinate (C) circle (1pt);\n",
    "\\draw [dashed] ($(C)!0.6!90:(A)$) coordinate (D)\n",
    "  -- ($(D)!2.2!(C)$);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3ecefb34-fd2c-40c5-850a-51874e686acb",
   "metadata": {},
   "source": [
    "从 A 点出发，向 B 顺时针偏转 90 度的方向，前进 1cm：`($(A)!1cm!90:(B)$)`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "1f2f13de-db75-482d-bed7-9488be7ff008",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANMAAACCCAQAAAB23DUAAAAAIGNIUk0AAHomAACAhAAA+gAAAIDoAAB1MAAA6mAAADqYAAAXcJy6UTwAAAACYktHRAD/h4/MvwAAAAlwSFlzAAAXEQAAFxEByibzPwAAAAd0SU1FB+kBBxQFL5ChM1MAAAg7SURBVHja7Z2/jttGEMZ/axhIzc41DaQOwEO6dLw+DQ0/ge4RqEegqtS6Jm0gNempR5DgLp0EGEhNlS43hf6REkktKS65VGZU+HwnCeR+OzPffDu7VBox9+2DDIHAJCYwCUxiApOYwPSk9lGG4GDKY8KOgJ1+d/DqpG4CUAERM70HNSfgVe8l6LloMd4RmiUBE8lNrlp0/HcPeJKb3LS3808hsJLcZOtGfL07UgHaZxYVkPKmlwKTLQKwImaKj4cPetrym7Z80Rsh5Fb8iFBPQXmkTPW7ythTgElN8Ks+nQdUhSxdBOk5ctOEGQABnn4H3rge6g27is8Ww2N4/CaByYK9H3ORf0j9t5nF2EM81+qlJyLkenf2hUcD1t7Ve3wephew5lWvyrheTW7a69kY7u4ZKETIRu8JQa+Oherqyi8Mc5MK3CQQgB75iwmaREPKVqPBI37sm1x8jd+bVuzYqIQpiYrZEbRma3t3s9MT5CblEbDRe1BBA1Y3rnuUhYwxmCjkApOYwCQwiY2+vFXJ3bf8xCe+y9AO603x3SXoT3yVge3QJgTNa3VNeOcdIVojr45UlogMX3KT22Wtz5yp3glMbtuclX6XziK3fSkm4HMrpifWG0gByan/VoKeywFvdlxBE5ic9aUELksyEvTcBCkk5uXSKiDe5CJIHnNm+ZUzgcnNrLQr9u1K0HPPlyaEvBR/J97kGkg+CVO9E5hcD3ir222lEvTc8qWz7iAwuQtSTneQoOc2DS/dqdjGm2Yqq/27J0PeyuK87vA4TH/xTcbUtu7wOEzf9EqGtfOAtyjqDpKb3KThm7r9wsL0HNUdxJtGoTsITKPQHSToueVLSbnuIDC5pTvEJueNSdBzVHcQmEahO0jQG4XuIN7kSsCr1R0EJldo+Mb8nDIJesP40l3dQbxpeJAMdAeByYWAt2p2jLYEvf59yUh3EJiGBclQd5CgNywNN9QdBKYhzVh3kKA3nC810B3Em4YLeA10B4FpOBq+aXs++kcXZpne/w98Kac7qAWr47GmATujZwh0f3xHw++LyYie/hAOn4xJbgxPr8R41GtOS0ltw0RAxpyMOd5Tw5SyyP1vTUJCgt/AOSr+4LHtAaY1cw0+a7bdeqlTICVk+WlYNq71rw81DN96xlAJHlPQO/3CktTgdLGx6g5fHsu/Hyu/ekPQQxVxlk30VC1ZqJC35zpUt1x3UCEBHvvzgyPaUYjjyd4Wgx4e2+sEikdC1vYccWcD3vo6757oBBO2Zvmpin351mGas648mCx9FkJBiCa4/W0uN6ctc5MKco8HsRUKIqLcg+Ly3r3kM3u2Knpe3SEXAleEym+nQkS2H+9xezxF4Sb2+gsz5mquxr6hrVR3UKEKC7/wW+SmQ8A7cn1LQa/8m28KwpGTdCZkZZmH7HJ6J4nZeH4YIODFBHy5O31OJP2PkQa86n6HTe6Rej57k9Wna0Lu4Z2rl5CdSrp++LUKiHkzriL+5Xf12yhJenW/w+b0wFblGT8mr5YEWgh6B93BUEjShHjMx0fSr3WHm79OCIlYd6HpWYCJhK0Z2c7XVWMj6YcJducdCZOHNT1SUjQZ6U0J+gBMhOafJsnXVXikY1HS8cy95GHptfuFDDy+84OFiVcQ3ZaFxONQ0st0hzHBNGdt5hV45dmIwH2SXq47jAYmIrLD5ROTkdTNNxbVdRWJu08APE4wC1fXE0xF/zh6RVCznuvXztct6+5nbCfDuWi+luQSTDe6Q5VX4BsFRSdJepXuMBKYiMuqiKNX+Dd11cI4iDpF0k0mmMMwEVRdPh6LoleY11XukfRiv8P4YDrrDnjHVo1cswYR2Ymk3y8LS/3UCZLeZII5CNPl8vFO/I4YffEC/INXlK3nGvmqAyS9+QRzCqa87kByAQddXLslRrMuX881mgqDknQbukOPMBX9gxh9ykPowxPUr7yiNckelqQXha3xwVTV7xCgTXXyBlNiIJJuR3foDaaL7lBSBK5tpNshSHqVsDUSmMovn5CE1B4365+k29Id+oKpst8Bn4XNOf8oScczpwP2dIdeYCrXHQqeZjHpPkbSWaDN6IhN3aEHmKp1h5yvabs32JakH4Rfs+5bm7pDHzCV9juwvtTppPYrnTYk/eIf9+mIXd3BOkxVl49G53oAtf1w0Zyk5yfYUXGMhtEdLMNU3e/A4qxHeNcqRMdelLYj6bcTrIqO2NcdrMJUp8vhMWdCSMjaHtO73T5nStLL/aO8+9a+7mAXpnn95RMQk9is2UnKdjncJ+m1E+yKjvShO1iEiaifKqJ2GkQVve93SHq9fxBc6Eg/uoM1mIa4/NvBri2sK0l6tbCVL3oP99eP7mAPpt6qiJqax6/f71FO0k0nGBEZ/7DvP2J0BlO97tBTwIvrZaoqkm7uH/zCjyGW9TuC6b7u0E/Auw/TLUlvMsGGihhdwbQeusXx3va5KiW9iS7Xp+5QfHVyZpFK8JgNuumr4fY5vedVxcxVyK8sjU4NanmuZEf3p5t/4LW4v02FpLwMu1FMhYS5g0d2LE22z6mAP/mZr/pvo/3CKau2J3g9bI8GvXb9QJbjeNoko5mJwH3rDh3npnu6g+swVXXf9t/vYBGm4XWH2zVjSrfPPaKkD1+4PwSTC7pDp+0uFVvkhtAduoRpcN2hj3aXvvodLME0vO5gqd2lsEWuv34HKzC5oDtYbHcJ3IoY7WFau7u1srt2l+F0h05gcuXyLbe7RH32O3QO09BVRE+EYo4zm7HbfCByTXewR9JduZI2mt4G9AtiPVobhdxHQHIepg1L2yfuiV3bfzlF5WfattarAAAAE3RFWHRwZGY6VmVyc2lvbgBQREYtMS41UzZawQAAAABJRU5ErkJggg==\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%tikz -l calc,angles\n",
    "\\draw (0, 0) coordinate (A)\n",
    "-- (55:1.2) coordinate (B) node [midway,right] {3}\n",
    "-- ($(B)!1.6cm!90:(A)$) coordinate (C) node [midway,below] {4}\n",
    "-- ($(C)!2cm!-90:(B)$) coordinate (D) node [midway,right] {5}\n",
    "-| (A) coordinate [midway] (E);\n",
    "\n",
    "\\draw ($(A)!0.5!(E)$) node [left] {4} ($(D)!0.5!(E)$) node [above] {$x=?$};\n",
    "\\draw pic [draw,angle radius=3mm] {right angle=A--B--C};\n",
    "\\draw pic [draw,angle radius=3mm] {right angle=B--C--D};\n",
    "\\draw pic [draw,angle radius=3mm] {right angle=A--E--D};"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "83a331fb-41e1-4156-ad2a-1ae25647e941",
   "metadata": {},
   "source": [
    "### 垂线\n",
    "\n",
    "`($(A)!(P)!(B)$)`"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d72bca85-1a3e-47b6-8dda-766d5e2f63e2",
   "metadata": {},
   "source": [
    "Altitude lengths are 210, 170, 357. What is the perimeter?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "02fd0f54-55cd-422f-9db4-4c8a0b663322",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%tikz -l calc\n",
    "\n",
    "\\draw [thick] (0, 0) coordinate (B) node [below] {$B$} -- (4, 0) coordinate (C) node [below] {$C$}\n",
    "  -- (1.6, 2.3) coordinate (A)  node [above] {$A$} --cycle;\n",
    "\\draw [blue] (A) -- ($(B)!(A)!(C)$) coordinate (D) node [below] {$D$} node [near end,left=-2pt] {170}\n",
    "  (B) -- ($(A)!(B)!(C)$) coordinate (E) node [right] {$E$} node [midway, above left] {210}\n",
    "  (C) -- ($(A)!(C)!(B)$) coordinate (F) node [left] {$F$} node [midway, left=3pt] {357};"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6c02fbe3-f7de-49b8-aba0-452a691ebe75",
   "metadata": {},
   "source": [
    "### 坐标分量\n",
    "\n",
    "直角三角形与正方形，求蓝线长度。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "5176cab4-3956-4b82-b1b3-c43eaac38faa",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%tikz -l calc\n",
    "\n",
    "\\draw (62:1) coordinate (B1)\n",
    "  (0, 0) coordinate (B)\n",
    "  -- (4, 0) coordinate (C) node [midway, below] {17}\n",
    "  -- ($(B)!(C)!(B1)$) coordinate (A) -- cycle node [midway,left] {8};\n",
    "\n",
    "\\draw let \\p1 = (A) in (B) rectangle (\\y1, \\y1) coordinate (D);\n",
    "\\draw [blue] (C) -- (D) node [midway,above] {?};"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e3368435-7992-4c56-bba9-0be9d4cdbae9",
   "metadata": {},
   "source": [
    "### 直线的交点"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "3f8a586e-f835-48b1-9af1-92c99e59547b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%tikz -l calc,angles\n",
    "\n",
    "\\draw (0, 0) coordinate (A) node [below] {$A$} -- (2.5, 0) coordinate (B) node [below] {$B$} \n",
    "-- (2.5, 2.2) coordinate (C) node [above] {$C$} -- cycle;\n",
    "\\draw [blue] let\n",
    "    \\p1=($(A)!1cm!(B)$),\n",
    "    \\p2=($(A)!1cm!(C)$),\n",
    "    \\p3=($(\\p1)+(\\p2)-(A)$)\n",
    "  in (A) -- (intersection of A--\\p3 and B--C) coordinate (D) node [right] {$D$};\n",
    "\\draw pic [draw, angle radius=6mm] {angle=B--A--D};\n",
    "\\draw pic [draw, angle radius=7mm,double] {angle=D--A--C};\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d3dbb897-7834-4a77-b775-6a80085e2eb4",
   "metadata": {},
   "source": [
    "### 圆、扇型、圆弧"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "a0b47732-d4bf-4be3-8b59-a584bfa4428e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%tikz\n",
    "\n",
    "\\draw (0, 1) circle (1);\n",
    "\\draw (2, 0) arc (0:180:2) -- cycle;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "c7a969bb-e089-457d-9ecf-906512ea3884",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%tikz\n",
    "\n",
    "\\draw (2, 0) coordinate (B) arc (0:90:2) |- (B);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "39527321-b2b3-4a38-8e71-8f429b16bcc1",
   "metadata": {},
   "source": [
    "### 曲线交点"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "01a21da3-e582-4fe4-a8d0-e2b50813162b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%tikz -l angles,intersections,quotes\n",
    "\n",
    "\\draw [name path=semi] (2, 0) coordinate (B) arc (0:180:2) coordinate (A) -- cycle;\n",
    "\\path [name path=line1] (-0.5, 0) coordinate (C) -- ++(60:2.3);\n",
    "\\path [name path=line2] (C) -- ++(120:2);\n",
    "\\draw [name intersections={of=semi and line1, by=E}] (C) -- (E);\n",
    "\\draw [name intersections={of=semi and line2, by=D}] (C) -- (D);\n",
    "\\draw [red] (D) -- (E) node [midway,below] {3};\n",
    "\\draw pic [draw, \"$\\alpha$\"] {angle=B--C--E};\n",
    "\\draw pic [draw, \"$\\alpha$\"] {angle=D--C--A};\n",
    "\\draw (1, 0) node [above] {$\\alpha=60^\\circ$} node [below] {$r=?$};"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7d24c4a6-d423-424a-b450-857a27f05bb2",
   "metadata": {},
   "source": [
    "### 三角函数计算\n",
    "\n",
    "直角三角形内接半圆，求半径。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "15a19a3d-8738-4f80-8459-4000343deb29",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%tikz -l math\n",
    "\n",
    "\\draw (0,0) -- (4, 0) node [midway,below] {$4$}-- (0, 3)\n",
    "  -- cycle node [midway,left] {$3$};\n",
    "\\tikzmath {\\r=(3*4)/(3+4);\\a=atan(3/4);}\n",
    "\\draw (0, \\r) arc (180:360-\\a:\\r)\n",
    "  (0, \\r) arc (180:180-\\a:\\r)\n",
    "  (\\r,\\r) node [above right] {$r=?$};"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "10216c0d-6ae5-4dd4-b5ee-f72dd4d7974d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%tikz -l math\n",
    "\n",
    "\\tikzmath {\\r=(3*4)/(3+4); \\a=atan(4/3); \\x=(3-\\r)*5/3;}\n",
    "\n",
    "\\draw (0, 0) -- (\\a:3) node [midway, left] {6} -- (5, 0) node [midway, above] {8} -- cycle;\n",
    "\\draw (\\x+\\r, 0) arc (0:180:\\r);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1049fe6e-ebb8-44ee-a965-d7ecd9b5fc86",
   "metadata": {},
   "source": [
    "## 例题"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dd76483a-47a7-43a2-8cb5-8e52511a15e6",
   "metadata": {},
   "source": [
    "$EF \\perp CD$, $|EF|=|DF|=|CF|$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "17b7d3f8-c82f-4b2c-85d7-835d1624e31a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%tikz -l math,calc,angles\n",
    "\n",
    "\\draw (2, 0) coordinate (C) node [right] {$C$} -- (0, 0) node [left] {$B$}\n",
    "  -- (0, 3.5) node [left] {$A$} -- (1.5, 3.5) node [right] {$D$} -- cycle\n",
    "node [near start] {/} node [near end] {/}\n",
    "coordinate [midway] (F);\n",
    "\n",
    "\\draw (0, 1.5) coordinate (E) node [left] {$E$}\n",
    "  -- (F) node [right] {$F$} node [midway] {/} node [midway,below] {$x=?$};\n",
    "\\draw pic [draw,angle radius=3mm] {right angle=C--F--E};\n",
    "\\draw (0, 0.7) node [left] {6};\n",
    "\\draw (0, 2.5) node [left] {8};"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "4b58c195-fd34-4518-8cb5-590cadcb798f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHsAAACXCAQAAACfSOwNAAAAIGNIUk0AAHomAACAhAAA+gAAAIDoAAB1MAAA6mAAADqYAAAXcJy6UTwAAAACYktHRAD/h4/MvwAAAAlwSFlzAAAXEQAAFxEByibzPwAAAAd0SU1FB+kBCBEnCbu2IjMAAAjTSURBVHja7Z27buvYFYa/bUyKAMEAe4pUqegu5dD1pOF5g9CPQD+C3KWlnyCgmnRTSEWqVFSdIpDyAAHIMsE01EwxSBNkTSGKIiXeRG5eJIsGDmyLR/Svvf61/r0upBI+4/EVgPoz8BOwASCW+N5hKzn+8wMJP/A/bHT62o49e2JiYtncJ+xnPFw0azayBqWxAXDQWNho9uzYAZt7sIYUtigAZePhsmfNWnaF0zQ2VvoB2MCGDZviOTcLO/2Vi4vLjjXr8nVVNg42DrBhd5PwBQERil9oPLYIKzz0+avZWTYLQoSEFQusqvPm91UBO33RwiciIcCtfRMHnxAhuhXwtbCzNQ1IiPCxa8/TeIRNFnIzsNMTXVYI26b1bGshNwK7PePPLMS6edjXrmdqISHOHcC+kvEWAcncoHeGfQ3j0fgkhPNhe0/YV8T4A/QI705gt2f8fKAbg92W8Wg8IrbTct0w7HaMR+MjBNOJmkFgt2E8FiEJizuDfcF4p9QqJjL3gWE3abbU3P2xzX0U2BeM12cfSkgyrncfEfY54wu/90hYjbfmvWGj8QkIyzYdOIQ1jP+Zv5x4jcWWqF7ozkecalZYApqApPhHo4nKYacwhe+LjMdHxvHtfWFnsRdNQlR4zWdbA9tnlZp3jvE44xh7X9gJSbZWKyRntDYuYQ3snERFszjFeMLhjf2pZwYyrvgeXFnXJGxtNNnrspcPeeGZmAURMf9gqxYTZE47rfwWyVZ+gSVUrzYBQWU6MiDh3/yXvw1n7CZ3YIJ/SiQLtbCThlysx18RfhwKuDnYIduTu8p+F1YIl6jFO/r8ZyiWPxkqrvhovqTfL1g2nO6ybvGmDn9iQ6jsgQq9vUF7aHnJnBX1pUGlcXlpfE8bm7Us1Z5QvZquuBqArVxseQNQFqDRys/WK1Y+sSzP1nrXomrmspQ9yLuKCdWbLGflybGPTD7674LjCEs9fgslRnRyeniI2a1KScXzqpW2WWU81TjyfFZO3ciXs/9hEfHcVCFXLoF8k/vZYcVS3mey2kSFH8OCZz/UQsOTNeRFaaPo9S+sKqmK9bPfeBZFac0GJ7kMXNgk5x/Grey3bZJmCYJXHtexTaUjxoYdtDFUwiqnh2vGuY0Ne0tC0Fg1k+rSEh7SX7mNz223qQTMonq7miWg7BuDnWXUDsmFkhx6s9O7zOTcBOzMmI/JBbfgtqSF0wvY9tmdTQj7pPLSAoKdxvVW0ZlVH+CTw75g/I/tquBott3ly0xgZ4z/Oz9fQZLOUXxGsMtFaYONdHRts4KNvjYm43dj+Lxge22STRcCKLh12OH1NZFuDJ+TS6sVpWYZPifYDaLUJMOfmM/htcqnlh0fgD9RVaS3WpMeqssmuaYZZD6rnWZKO6bGdiwJeg1N9EwfaxZoLHYsT4lCpTmW8pYVgxgJb7LudeWIdesko+F2vNIyP/rYlMMCKdPcuCS9r+20jwSmYZeW+fFPYJFTrayrKK3ZlYXTwC4t87M4NWcgl0rselFaI13cKWBvc7D9fHdDzl8HJkRpZeyP2sSDAQNYvsxfnxzoIkprrupPCDtf5k8djk9Y1l7bVZT2cWzDwQ4vXRcWK8Jz4N1FaaVbDSeCXaWS0STnH4fptnp0s2IbBDZe9R6YsBi5+4nSyg89HF2c5sv8ygJQWxUpnTvFNiVKK44ljnLqTjAOW9lH0ICbwbTIw94V9l3GR+IlZsliRE1eXuZXK5aH7hOlSdjJS1X53tiH39A8YBp2hJX7Me1lUBqfHTHgs+f1ZNQqYG+wRyH/lwSQWd3wO7AaO3DQxbsAKE3CyzBD7w3rPWWawZwovbaWPi3scMju8Tr1N5qRX+92DFwhJK7g94SFXsOitFSfV3TKTJlL654pbRu/N+wz7TCsXLnCs1tDwwaW5bAn47by0dVxdWj/8TShiY9wnyaJ2ZSt90SwlcvXuMob4VJrvNkYuQr4Pz+lN69aD3l/rgolOFGDVpopHeNGJWXJ6Glgn4nSqqFXY+1AyTxgl4jSqqFXQ13N7uSw67Ry/Zh7jyTTanrYjaI0HYHzjV3RPjfzKWB3bqTvVaRyJtXkLUWpy95o+mGDM61caZcpdQzr9R32pHG7TU3SbHnomI2fcnqg3XTnoqwGbpLdYxt5O+N1WRq/cpHdY652u/L9EOUhAa8wpTYqbK+N8bYbkOsikqYycreliQ+QdZGYWLkTBDBl4TSnFpSLlmGSTZtTEBtztTU7tmql3FEjdnnsHjmAnQ3EDC1Kz9PH0248K0fgBi4PZRFisvJA+QgcK3P7rjrJMnGHcXEEzrwovUhveJPXwAq7MhcXzb/4Wn4/aG6eQzV9Fo3VspN3eeaNmN+pSPnKGuxS9gzq220Zb9CXb2c5/lbG+CEE6ixht4vxHUOYPXPYTTG+YyOuM7fJoHJ3t5Y3nvnAIlJb5RXa+q4/4sOue/awAWQvS3nlmTUeSaOqr4fNzcA+bh3lQ154IcZXifI73llL3xjssxivCVWkFlfHeGuGcbtTjA/bx/jjnVdvbrVLGL/BI1HBVYy/3dUujfFND4BI99x3AjsX42sf4XPUaXcFO2N8WM34O4WdU/URcvkAiEOG5U5h5xifFBl/GMm7YU/eMsZ/wxuabTHGH5/194X7Pn7Dd/yBb/kn3/JFNgfYIZ/j+DW/5Vf8UXbqcz6+9IkH7AfsB+wH7Ls5vprnn6U0HjH2xd2ujSmZuYpKnTZFb4foOp6nkS/QacveGhvv83D7mCnZH5N+n4Hbp4mhw1Ofb4Xbp+xGn1oGdtvbx1z7NYgmVx7g8ZY+sN6W147vE/E6zJjzMEZuybtyWfEqOxWdc1N5VOa28yPsyjl/5vusYatD95mVPoX9nfOZux1VU7zFhmuHj+EyEIPwGgvpy0pzTxoYJW5LbMgD74cLFQOlGVSALS+VDq+K23sZzqxHidt1LZQtua3soRzaQHHbCLO94h24bkGTW4aYvb81bltD3mpitrAf2ZUH7AfsB+wH7AfsB+wBjl8AK4jVj3vvNYkAAAATdEVYdHBkZjpWZXJzaW9uAFBERi0xLjVTNlrBAAAAAElFTkSuQmCC\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%tikz -l calc,math,angles\n",
    "\n",
    "\\draw (0, 0) coordinate (O) -- (2, 0) coordinate (D) arc (0:90:2) coordinate (A) -- cycle;\n",
    "\\draw (A) -- (50:2) coordinate (B) node [midway, below] {24};\n",
    "\\draw ($(B)!1cm!90:(A)$) coordinate (B1);\n",
    "\\draw (D) -- ($(B1)!(D)!(B)$) coordinate (C) -- (B) node [midway,left] {23};\n",
    "\\draw pic [draw,angle radius=3mm] {right angle=A--B--C};\n",
    "\\draw pic [draw,angle radius=3mm] {right angle=D--C--B};\n",
    "\\draw (1, 0) node [below] {$r=?$};\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "02fc7e4e-60aa-41f6-99c4-6cc5ece4bb16",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%tikz -l angles,calc\n",
    "\n",
    "\\coordinate (O) at (0, 0);\n",
    "\\coordinate (T) at (0, 2);\n",
    "\\draw (-2, 0) coordinate (A) arc (180:90:2) |- (A);\n",
    "\\draw (110:2) coordinate (B) -- (A) node [midway,right] {24};\n",
    "\\draw ($(B)!1cm!90:(A)$) coordinate (B2)\n",
    "  (B) -- (intersection of B--B2 and O--T) coordinate (C) node [midway,below] {7};\n",
    "\\draw pic [draw, angle radius=3mm] {right angle=C--B--A};\n",
    "\\draw [thick,blue] (O) -- (C) node [midway, right] {$x=\\;?$};"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "ed032065-0b30-4f4e-984b-426e21e76f41",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%tikz -l calc,angles,intersections\n",
    "\n",
    "\\draw [name path=quarter] (0, 0) -- (2, 0) coordinate (C) node [midway,below] {$r=10$} arc (0:90:2) -- cycle;\n",
    "\\draw (0, 0.5) node {=} (0, 1.5) node {=};\n",
    "\\coordinate (A) at (0, 1);\n",
    "\\coordinate (D) at (-2, 0);\n",
    "\\path [name path=line1] (A) -- ($(A)!-1!(D)$);\n",
    "\\path [name intersections={of=line1 and quarter, by=B}];\n",
    "\\draw (A) -- (B) -- (C) node [midway,left] {$x$} -- cycle;\n",
    "\\draw pic [draw,angle radius=3mm] {right angle=A--B--C};"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "68dfce16-7d77-4d46-b231-8f64a430ace4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAASwAAADGCAQAAADC3HWuAAAAIGNIUk0AAHomAACAhAAA+gAAAIDoAAB1MAAA6mAAADqYAAAXcJy6UTwAAAACYktHRAD/h4/MvwAAAAlwSFlzAAAXEQAAFxEByibzPwAAAAd0SU1FB+kBCAYzGgjPAU0AAAwgSURBVHja7V2/j+NaGT13tCUUnidRsQWe1yFR4O2h8CBaCo9E/SRP8XqyoqL0VHRPykj0yPMnOA29UyKqhB8FFXLQe0tBw0eRZDbJOP55bV/fe5xiNZsd787dk+8cn+879yoBL176r3fj/5UqgGfByq1lR/hU/C+PXbGUhwKr2a/bD3CLnxBaJlWsCFu5t+ATmWOJBwLo2nUz+t8Y4MWKlXtAqBYEkDlUWOBBVlYsXYQUH2RNEBkALBUgk1trFi9BhA9UWiZQYYSVPYsnH7HDkiAyAVihTcCi0jKECpWPDe5ka9UCUmkZULFCrO2CFSAveEKqPEJpWmCt7FtCKi0TgPVi5SpSaU0JLBUCdmoR2eIRiQoIp2kqlq31ikprYmBFsPjZiUprImApH769FYtKa7qKFWFld/ODSmsaYAU2Wg1UWkZULPuXs4nSUp5K7F+JkVo6KkIidy58UpWPHE/ydOW9GB4CBKJYsXRZDSs4cVUqrR1W8ujGShBYIyot2YkzqzAKsFQAX17gzEVPa6yKFeLFsXV13tMaC1iOzSvR0xoBWMpzsGI572mNUbEibG0b76PSMgFYgXv1ikprnIq1cnNxXVZagwNLBYA4CiyXldbwFcvZelWltOyH2vDACt0G1rnSUpnKEAPIVGZ3K3rgJrSNScIOq+Bg9nDoimVhkpBKywxguU6EFUqLwOoDrBfC6lJpEVh9tYW1ScIONcsxT2vYisV65azSGhZYEVivHFVaAwLLgSQhldYkFcv6JCGV1jTACmg1uKu0BnTelXCnuysrk2Mrlu8RP1jFUhG2hJW7Sms4KqTn7rTSIrCotOaksVSA3P4YOZXW+BWLnnszpfVEYLUFFoV7vdL6E35jq9IahAqVh4LjfbVrlCLACoGdp/EMU7EcTRK2gFWIDYA7ebC1ezgMsAIqrEpYJcjwLPeyg72elgzwQoFQwFfp2njIztcHEQSBbT/nABXL7SRhQwpc2e5pDUGFEa3RBhR4yho2zmkNUOxzxKS8ego8e9dHgYVVP6/2G/oQ+ATSxaqEKJDBq/gTlikt/VTIJGFjCjxjDsuUlnaDVKXYykeC6cIIfWjyOGNT93CIikUPq/IpsOKyydPSryWoqF5XI4EgafUd1iitd6xXg1LgfTtHT17UE1JlQfdQNxUySdiFAm30tLRbDR4psD0F2udp6b3ZAhmN0Coj1B2lpZcKnU8SqhDbbhRom6el1cdyPUmoEizwN/mRlnvN3NPSWLHcThIqT2WI8Sv8T9PeonP3tDSqiyWW7AUigOiZRpu30tJ5qw0iPgUKsECh59kYCTZzfcrWprFcTRKW9QJVCk/u3VZa+jSWk577FSP0Eb42pfUL9TunNRYyuwbV+hmhupQWInw3T6WlzxZ0bLyvzgjVobSQoEA8T6Wl6zYxNpwIvfgzaZ8+BHzkyPe1CjlSV4G17N4bs7UXCA+brquCCAWWR+DOsXvIJOGAvcCuSmtPgfP2tPTcJHBlvK8JBfZTWqcUOGdPSxc1pKRAHUrrnALfxOpS14DlQJKw6zhMG6X1lgLnO6fFJOEgFNheaV2jwLkqLT1WQ04K7Ke0qihwnkpLxy1Sm60GHROhdUqrmgLnqbT0WA0BKbCr0mpCgXNUWlqWnhTYVWk1pcD5KS0di78kBXZTWm0ocG5Kq/8NNjZaDboo8LrSakuBc1NaOqwGjxTYVml1ocB5Ka3+5T0jBbZVWl0pcE5Kq39xX5ACW34UpTsFzkdp9f/2gBTY2lDWBAeTlVbfcrwhBU74bzZYad30DFCsrA5FmJ5XMPncw55WQ0QKnPzfbqTS6pErtCNJ2GaPUEN/AiOzh32o0IIk4TwpcA67PPQD1sy3AGmyTTaV1sgaa+5Jwjk+Bc5JaXWvWAadSahClTlIgaflwbidS7sDy5gzCZXXdkntoEDDlVaP8b7QGBrIm3cs7aJAc7uHHSuWOWcSqgBrNK48tlHgWYkwaufSrlRozpmEkTSmZCsp0FCl1RVYhjRz1ALPTZWYyhDj3vIDpIxRWp2ApXwzNt5WAdDsydRmCjTT0+pWsUw5kzCSJ1KgmUqrK7BMqFeNaNARCjROaXUF1uQeVjMadIUCzVNaHaYbVIhUbicHVojw9YsFtnjBVp7fUOACTy6e96oipBOfEmJDkhDy1iC11widR/ewCxXO4ExCRynQJKU19yQhMmQQFMhOzoaY8USoLRPxrTWWWiDUc+rCQLVq9hOh4ykt5eEo8Z+120d2JQmHzwXao7TgIdm/hwVEd3rBqiQhKbBN9hDJZzhBdG+ed9O6uBp6JqF7RmhvT2sHwH/9ypuUCk09k5AU2GdOCwFE9/+rFUlCUmA/Twspct0fylZPhSYmCfkU2GiVrmQPVYgQAbb4qL1B33KnlJQUaJunBR/pAJvMtTQjF6RA+5QWPBS6nwpnmyR0vReo2dPKdDtZbewGo5KErvcC+3cPVa42ZwOBWudO2wDLnCShQxOhA3paAfwz90qvPzm3JCEpUJfSQnpcRXj6nffGdoMKkBkx3pdijQfWqh7VPsIH2QHKQ4I1tgAS7HSvaXNgJfCn3oXJ3YnQoTwtFSCEh5cB2nQtmpkxKZBzWprthqnPJKQROrddHpo+FU6aJORToGaWGiF72FBjqRTbabQNe4Hjdg/H9rEmShLSCB3J0xqgLDZUOOwFUmm1eb0ztV4dKPCetWo4paWekKoPwyjXZlQ4epKQFDgKtIbMHpqYJCQFzt/TajbeN+KZhDRC7VBaTahwxE3WSIHWeFomJQlJgaZlDwekwrHOJCQF2qW06qlwlN37SIGT0uEQO5eakCQkBZqfPdQ86Dd8kpC9QDu7h3VUOLDnTgq0tXtYD6wBPXeOw1istKZKEvIp0G6ldVPTIxwoSUgKtL17eFPjuQ+isEiBDiitsZOEpEA3uoc3VVbD3fcQq5gUyO6h1or1/huRXBK53SHRI+FphLrTPax464t/JJKLSCGJePI5kE0KZPewB7DgQ/BbZKFkn8G17Fq5mAt0TWldfyPebxMBH8kvv92DayEoulAZKdA9T+v6G+kpFBD/+ruNiGwklnZtaVKgm0qrymoILgCSxLIHV9R482ZSoKtKq1WSED7SPbgW0mQDZ1Kgu0rrOiSW1yD3xV9zEVnKZU0jBVJp1QNrc33TInjIlkdoxaRAKq3GwKpPEmK5FJFMPCmDFimQSqv8NxskCREnIpK/gRYpkErrOrAanUmI4OtPIrkEJ9AiBVJpVQGrIT4RfP2pkEKCw+bzpEAqrQpgtUkSIgykkEK+/Dd+Tgqk0qoG1tmZhHW1C3EgheSC/5ICqbSqgXVo2cDHEgkiLI8aqhw4WMSSyftvuPRUWhXAQgA5/LrZ4xPenuLgl1oLHjJ8IgVSaZ2/bsqThMpHhuf9tvKywxoxgPjtBPxhIvQ9J0I5EV8XptgnCRfw8Pz6e2v4ygMuww/q9wxFMHvYaDT5mCSEnJY8JBAszof84CHDf/BPCnYqrSZUGGErWxUA2Jag9ZICf4g/I+HnmNnDeircJwlLQqryVJoL/IhY+VxsKq06KjxYnMg+O+iIkEEE8OG/7QUiazr0x5dLntal1VCcKKgYIWIkCOEhR4RlWS8QIYQ6i0qrGljJmWQPEB6/FR7Ca73A0+rGFz2tMmDl1eN95b1AhChYs9g9vAqsqjMJq8dhWLOotKqAFV87cLpuHIY1i0qrClhpGXhOKfD6cwBy1iwqrWsGaXi1F3iHrVqqBL5aHnefudiN5BkB7R16WiU+VlmS8EiB9ZMOx5kIvqi0LqjwMkn4mQLP1T8ypAIkl8zK6VEqrXIqPDuT8GyDtEaTDliTDNk9fEOF50nC86fA+kmHsorHl9ue1s1rvVrta5DyVIYY98cz6+snHV4rVsjPLue0LqcbDmcSluwRWjPp8Hqt9hTJy1FoXexcelKxyrfJlt3p6V8q2isp5V8Oy8gOW6osKq0zjYUIm4pe4JVJh5LYGE1SKq2D0np3sEb/gg3WuCubXpcd7lUADy8HFXaPQB5LIMvJd+eVlnpEqlayxv5YOfUtvo8/4g897/sV0PsevOZ+fYWf4cey2wPr7/gXit63vAU03IXX3K8v8dMDsHjx0n3dcAl4EVi8CCxeBBYvXgQWr3lc/wfXGeYmuEdR5QAAABN0RVh0cGRmOlZlcnNpb24AUERGLTEuNVM2WsEAAAAASUVORK5CYII=\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%tikz -l math,calc,angles,quotes\n",
    "\n",
    "\\tikzmath{\\a=atan(3/4);}\n",
    "\n",
    "\\draw (0, 0) coordinate (A) -- (5, 0) coordinate (B);\n",
    "\\draw (A) -- (\\a:4) coordinate (E) node [midway,above] {4} -- (B)  node [midway,right] {3};\n",
    "\\draw (E) -- ($(B)!4/3!(E)$) coordinate (C) node [midway,right] {1};\n",
    "\\draw (C) ++(-1, 0)coordinate (C1);\n",
    "\\coordinate (D1) at (2*\\a:1);\n",
    "\\draw (C) -- (intersection of C--C1 and A--D1) coordinate (D) -- (A);\n",
    "\\draw pic [draw] {right angle=A--E--B}\n",
    "  pic [draw,\"$\\alpha$\", angle radius=6mm] {angle=B--A--E}\n",
    "  pic [draw,\"$\\alpha$\", angle radius=7mm,double] {angle=E--A--D};"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "f5325ad7-c84c-40f4-878f-25ff76925092",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%tikz -l calc,math,angles,through,quotes\n",
    "\n",
    "\\tikzset{\n",
    "  angle bisector/.style n args={4}{\n",
    "    insert path={\n",
    "        let \\p1 = ($(#2)!1cm!(#1)$),\n",
    "        \\p2 = ($(#2)!1cm!(#3)$),\n",
    "        \\p3 = ($(\\p1)+(\\p2)-(#2)$) in\n",
    "        (intersection of #1--#3 and #2--\\p3) coordinate (#4)\n",
    "    }   \n",
    "  }\n",
    "}\n",
    "\n",
    "\\tikzmath{\\r=2.5;};\n",
    "\\draw (\\r, 0) coordinate (B) arc (0:180:2.5) coordinate (A) -- cycle;\n",
    "\\draw (A) -- (70:2.5) coordinate (C) -- (B)  ;\n",
    "\\draw [angle bisector={A}{B}{C}{B1}];\n",
    "\\draw [angle bisector={B}{A}{C}{A1}];\n",
    "\\coordinate (O) at (intersection of B--B1 and A--A1);\n",
    "\\draw ($(A)!(O)!(C)$) coordinate (E) node [above left] {$E$}\n",
    "  -- ($(B)!(O)!(C)$) coordinate (F) node [right] {$F$};\n",
    "\\node [draw] at (O) [circle through={(E)}] {}; \n",
    "\\draw (E) -- ($(A)!(O)!(B)$) coordinate (D) -- (F);\n",
    "\\draw pic [draw,\"?\"] {angle=F--D--E};"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.11.9"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}