三角形与内切圆几何作图探究
2021-07-30 15:35
本GeoGebra范例演示三角形及其内切圆的动态几何构造过程。通过Point、Vector、Segment、Polygon等命令构建三角形基本图形，利用Sequence生成点列，借助Intersect求解交点，并运用Angle命令标注三角形的各个内角（α、β、γ、δ）。DynamicCoordinates实现点的动态坐标控制，If条件语句用于逻辑判断与图形切换。该范例适合用于探究三角形内切圆的性质、角平分线交点以及三角形各元素之间的几何关系，是解析几何与平面几何教学的互动演示工具。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.656.0" app="classic" platform="w" id="762933da-685b-4110-92d2-98d23790006b" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="950" height="523" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0" orientation="1" />
</panes>
<views>
<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="true" location="1,1" size="200" window="0,0,400,400" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="950" window="0,0,400,400" />
<view id="2" visible="false" inframe="false" stylebar="true" location="3" size="266" tab="ALGEBRA" window="1205,367,470,828" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 | 1011 || 1008 1009 || 6" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="16" visible="false" inframe="false" stylebar="false" location="0" size="227" window="915,562,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="128" visible="false" inframe="false" stylebar="false" location="1" size="480" window="50,50,500,500" />
<view id="4097" visible="false" inframe="false" stylebar="true" location="1" size="150" window="547,217,1066,951" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 | 551 550 11 , 20 22 21 23 , 55 56 57 , 12 | 69 | 510 511 , 512 513 | 533 531 , 534 532 , 522 523 , 537 536 , 535 | 521 520 | 36 , 38 49 560 | 571 30 29 570 31 33 | 17 | 540 40 41 42 , 27 28 35 , 6 , 502" visible="false" inframe="false" stylebar="false" location="1" size="300" window="50,50,500,500" />
</views>
<toolbar show="false" items="0 39 73 62 | 1 501 67 , 5 19 , 72 75 76 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 14 68 | 30 29 54 32 31 33 | 25 17 26 60 52 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="false" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="24"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="950" height="523"/>
<coordSystem xZero="475.497" yZero="369.7989999999997" scale="55" yscale="54.999999999999986"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="0" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="1" grid="0"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<grid distX="1" distY="1" distTheta="0.08726646259971647"/>
</euclidianView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2" max="2" step="1"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="point" label="C2">
<show object="false" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="4.44550909090909" y="4.3054363636363595" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="point" label="A">
<show object="true" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-29" y="30"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<ggbscript onUpdate="SetValue(C1,A+uCopy)
SetValue(C2,B+vCopy)"/>
<coords x="1" y="0" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<caption val="P"/>
</element>
<command name="Point">
<input a0="Sequence[A + (i, 0), i, 1, 20, 1]"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="6" y="32"/>
<labelMode val="3"/>
<animation step="1" speed="1" type="1" playing="false"/>
<ggbscript onUpdate="SetValue(C2,B+vCopy)
SetValue(C1,A+uCopy)"/>
<coords x="6" y="0" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<caption val="Q"/>
</element>
<command name="Vector">
<input a0="B" a1="C2"/>
<output a0="v"/>
</command>
<element type="vector" label="v">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<ggbscript onUpdate="SetValue(vCopy, v)"/>
<coords x="-1.5544909090909096" y="4.3054363636363595" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint exp="B"/>
</element>
<command name="Sequence">
<input a0="(x(B) + (Length[v] * cos(t°)), y(B) + (Length[v] * sin(t°)))" a1="t" a2="0" a3="355" a4="5"/>
<output a0="UPoints" randomResult="{(10.577469198997253, 0), (10.560050546719335, 0.3989527279431336), (10.507927156347076, 0.7948691847323764), (10.421495718354183, 1.1847362070706697), (10.301414028172502, 1.5655866715098747), (10.148595979971628, 1.9345220760518114), (9.964204611372429, 2.2887345994986266), (9.749643252028495, 2.6255284706675224), (9.506544843440125, 2.9423404848372026), (9.236759511283513, 3.236759511283512), (8.942340484837203, 3.5065448434401256), (8.625528470667522, 3.7496432520284944), (8.288734599498628, 3.9642046113724274), (7.934522076051811, 4.148595979971628), (7.565586671509875, 4.301414028172502), (7.18473620707067, 4.4214957183541825), (6.794869184732377, 4.507927156347077), (6.398952727943134, 4.560050546719334), (6, 4.577469198997254), (5.601047272056866, 4.560050546719334), (5.205130815267624, 4.507927156347077), (4.81526379292933, 4.4214957183541825), (4.434413328490125, 4.301414028172503), (4.0654779239481895, 4.1485959799716285), (3.711265400501374, 3.9642046113724283), (3.374471529332477, 3.749643252028494), (3.057659515162797, 3.5065448434401256), (2.763240488716488, 3.2367595112835126), (2.493455156559875, 2.9423404848372035), (2.250356747971505, 2.6255284706675215), (2.0357953886275717, 2.2887345994986266), (1.8514040200283723, 1.9345220760518116), (1.698585971827498, 1.5655866715098756), (1.5785042816458184, 1.184736207070671), (1.492072843652923, 0.7948691847323762), (1.4399494532806658, 0.3989527279431337), (1.422530801002746, 0), (1.4399494532806658, -0.3989527279431325), (1.492072843652923, -0.7948691847323771), (1.5785042816458175, -1.18473620707067), (1.6985859718274972, -1.5655866715098745), (1.8514040200283715, -1.9345220760518105), (2.0357953886275726, -2.2887345994986275), (2.2503567479715056, -2.625528470667523), (2.4934551565598735, -2.9423404848372026), (2.763240488716487, -3.236759511283512), (3.0576595151627965, -3.506544843440125), (3.3744715293324763, -3.7496432520284935), (3.7112654005013708, -3.964204611372427), (4.06547792394819, -4.1485959799716285), (4.434413328490126, -4.301414028172503), (4.815263792929331, -4.4214957183541825), (5.205130815267624, -4.507927156347077), (5.601047272056866, -4.560050546719334), (6, -4.577469198997254), (6.398952727943132, -4.560050546719334), (6.794869184732375, -4.507927156347078), (7.184736207070667, -4.4214957183541825), (7.565586671509876, -4.301414028172502), (7.934522076051812, -4.148595979971628), (8.288734599498628, -3.9642046113724274), (8.625528470667522, -3.7496432520284944), (8.942340484837203, -3.5065448434401265), (9.23675951128351, -3.236759511283513), (9.506544843440125, -2.942340484837204), (9.749643252028493, -2.625528470667524), (9.964204611372427, -2.2887345994986292), (10.148595979971628, -1.9345220760518103), (10.301414028172502, -1.5655866715098743), (10.421495718354183, -1.1847362070706695), (10.507927156347076, -0.7948691847323767), (10.560050546719335, -0.39895272794313424)}"/>
</command>
<element type="list" label="UPoints">
<show object="false" label="true"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
</element>
<element type="point" label="C1">
<show object="false" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="2.7546" y="4.178163636363632" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="A" a1="C1"/>
<output a0="u"/>
</command>
<element type="vector" label="u">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<ggbscript onUpdate="SetValue(uCopy, u)"/>
<coords x="1.7546" y="4.178163636363632" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint exp="A"/>
</element>
<command name="Sequence">
<input a0="(x(A) + (Length[u] * cos(t°)), y(A) + (Length[u] * sin(t°)))" a1="t" a2="0" a3="355" a4="5"/>
<output a0="VPoints" randomResult="{(5.531630228982873, 0), (5.5143860078250215, 0.3949575984647426), (5.462784583286822, 0.7869093311232501), (5.3772186733668015, 1.1728722086230647), (5.258339486305562, 1.5499088204156553), (5.1070517644978555, 1.9151496902243497), (4.924506898856661, 2.2658151144914362), (4.712094166033113, 2.5992363176006714), (4.471430155182312, 2.912875762871166), (4.204346464743737, 3.2043464647437365), (3.9128757628711663, 3.471430155182312), (3.5992363176006714, 3.7120941660331126), (3.265815114491437, 3.9245068988566607), (2.9151496902243497, 4.1070517644978555), (2.5499088204156557, 4.258339486305562), (2.1728722086230645, 4.3772186733668015), (1.7869093311232507, 4.462784583286822), (1.3949575984647424, 4.5143860078250215), (1, 4.531630228982873), (0.6050424015352571, 4.5143860078250215), (0.21309066887675, 4.462784583286822), (-0.17287220862306518, 4.3772186733668015), (-0.5499088204156553, 4.258339486305562), (-0.9151496902243492, 4.1070517644978555), (-1.2658151144914358, 3.924506898856661), (-1.5992363176006719, 3.712094166033112), (-1.9128757628711663, 3.471430155182312), (-2.2043464647437365, 3.204346464743737), (-2.4714301551823117, 2.912875762871167), (-2.712094166033113, 2.599236317600671), (-2.924506898856661, 2.2658151144914362), (-3.1070517644978555, 1.9151496902243499), (-3.2583394863055624, 1.5499088204156561), (-3.3772186733668015, 1.1728722086230658), (-3.4627845832868216, 0.7869093311232499), (-3.5143860078250215, 0.3949575984647427), (-3.5316302289828734, 0), (-3.5143860078250215, -0.3949575984647416), (-3.4627845832868216, -0.7869093311232508), (-3.3772186733668015, -1.172872208623065), (-3.2583394863055624, -1.549908820415655), (-3.1070517644978555, -1.915149690224349), (-2.9245068988566607, -2.265815114491437), (-2.7120941660331126, -2.599236317600672), (-2.4714301551823126, -2.912875762871166), (-2.2043464647437374, -3.2043464647437365), (-1.9128757628711668, -3.4714301551823117), (-1.5992363176006728, -3.7120941660331117), (-1.265815114491439, -3.9245068988566603), (-0.9151496902243486, -4.1070517644978555), (-0.5499088204156546, -4.258339486305562), (-0.1728722086230643, -4.3772186733668015), (0.21309066887674988, -4.462784583286822), (0.6050424015352571, -4.5143860078250215), (1, -4.531630228982873), (1.3949575984647413, -4.5143860078250215), (1.7869093311232485, -4.462784583286822), (2.1728722086230627, -4.377218673366802), (2.5499088204156566, -4.258339486305562), (2.9151496902243506, -4.107051764497855), (3.265815114491437, -3.9245068988566607), (3.5992363176006714, -3.7120941660331126), (3.912875762871166, -3.4714301551823126), (4.204346464743736, -3.2043464647437374), (4.471430155182311, -2.9128757628711672), (4.712094166033111, -2.599236317600673), (4.924506898856659, -2.265815114491439), (5.1070517644978555, -1.9151496902243488), (5.258339486305562, -1.5499088204156548), (5.3772186733668015, -1.1728722086230645), (5.462784583286822, -0.7869093311232505), (5.5143860078250215, -0.39495759846474326)}"/>
</command>
<element type="list" label="VPoints">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
</element>
<element type="point" label="D">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-25" y="24"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<selectionAllowed val="false"/>
<coords x="-8" y="0" z="1"/>
<pointSize val="1"/>
<pointStyle val="0"/>
<caption val="A"/>
</element>
<expression label="E" exp="D + (7, 0)" type="point" />
<element type="point" label="E">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="4" y="21"/>
<labelMode val="3"/>
<selectionAllowed val="false"/>
<coords x="-1" y="0" z="1"/>
<pointSize val="1"/>
<pointStyle val="0"/>
<caption val="B"/>
</element>
<command name="Intersect">
<input a0="Ray[D, Rotate[D + (1, 0), 40°, D]]" a1="Ray[E, Rotate[E - (1, 0), -55°, E]]"/>
<output a0="F"/>
</command>
<element type="point" label="F">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-14" y="-9"/>
<labelMode val="3"/>
<selectionAllowed val="false"/>
<coords x="-3.5770094835540314" y="3.6857854916285424" z="0.9961946980917453"/>
<pointSize val="1"/>
<pointStyle val="0"/>
<caption val="C"/>
</element>
<command name="Segment">
<input a0="F" a1="D"/>
<output a0="k"/>
</command>
<element type="segment" label="k">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-10" y="-1"/>
<labelMode val="2"/>
<selectionAllowed val="false"/>
<coords x="3.699864593426191" y="-4.409326921327779" z="29.59891674740953"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="E" a1="F"/>
<output a0="j"/>
</command>
<element type="segment" label="j">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-11" y="4"/>
<labelMode val="2"/>
<selectionAllowed val="false"/>
<coords x="-3.699864593426191" y="-2.5906730786722214" z="-3.699864593426191"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="vector" label="uCopy">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<coords x="1.7546" y="4.178163636363632" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint exp="A"/>
</element>
<command name="DynamicCoordinates">
<input a0="C1" a1="x(ClosestPoint[VPoints, C1])" a2="y(ClosestPoint[VPoints, C1])"/>
<output a0="C"/>
</command>
<element type="point" label="C">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="2" y="1"/>
<labelMode val="3"/>
<animation step="1" speed="1" type="1" playing="false"/>
<ggbscript onUpdate="SetValue(C, If(Distance(C,G)<d, G, C))"/>
<coords x="2.9151496902243497" y="4.1070517644978555" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="R"/>
</element>
<command name="Segment">
<input a0="A" a1="C"/>
<output a0="f"/>
</command>
<element type="segment" label="f">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-29" y="-29"/>
<labelMode val="2"/>
<selectionAllowed val="false"/>
<coords x="-4.1070517644978555" y="1.9151496902243497" z="4.1070517644978555"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="A" a1="B"/>
<output a0="g"/>
</command>
<element type="segment" label="g">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-11" y="10"/>
<labelMode val="2"/>
<selectionAllowed val="false"/>
<coords x="0" y="5" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="vector" label="vCopy">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<coords x="-1.5544909090909096" y="4.3054363636363595" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint exp="B"/>
</element>
<command name="DynamicCoordinates">
<input a0="C2" a1="x(ClosestPoint[UPoints, C2])" a2="y(ClosestPoint[UPoints, C2])"/>
<output a0="C'"/>
</command>
<element type="point" label="C'">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="2" y="1"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<ggbscript onUpdate="SetValue(C', If(Distance(C',G)<d, G, C'))"/>
<coords x="4.434413328490125" y="4.301414028172503" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="R"/>
</element>
<command name="Segment">
<input a0="C'" a1="B"/>
<output a0="h"/>
</command>
<element type="segment" label="h">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="17" y="-30"/>
<labelMode val="2"/>
<selectionAllowed val="false"/>
<coords x="4.301414028172503" y="1.565586671509875" z="-25.80848416903502"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Intersect">
<input a0="Ray[A, C]" a1="Ray[B, C']"/>
<output a0="G"/>
</command>
<element type="point" label="G">
<show object="false" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="55.85705596318641" y="88.33065037120853" z="14.667797245280196"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<expression label="d" exp="0.3"/>
<element type="numeric" label="d">
<value val="0.3"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<animation speed="1" type="0" playing="false"/>
</element>
<command name="Polygon">
<input a0="{A, B, G}"/>
<output a0="t1"/>
</command>
<element type="polygon" label="t1">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<condition showObject="closed1 ≠ 0"/>
<objColor r="255" g="215" b="0" alpha="0.3100000023841858"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<command name="If">
<input a0="C ≟ G ∧ C' ≟ G" a1="true" a2="false"/>
<output a0="closed1"/>
</command>
<element type="boolean" label="closed1">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<expression label="text1" exp=""拖拉紅點，構作一個滿足以下條件的著色的△PQR：
∠P = ∠A, PQ = AB, ∠Q = ∠B。
△PQR 是否必定全等於 △ABC？ ""/>
<element type="text" label="text1">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<absoluteScreenLocation x="15" y="37"/>
</element>
<command name="Segment">
<input a0="D" a1="E"/>
<output a0="i"/>
</command>
<element type="segment" label="i">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-10" y="10"/>
<labelMode val="2"/>
<selectionAllowed val="false"/>
<coords x="0" y="7" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Polygon">
<input a0="D" a1="E" a2="F"/>
<output a0="t3" a1="l" a2="m" a3="n"/>
</command>
<element type="polygon" label="t3">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="215" b="0" alpha="0.25"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="l">
<show object="true" label="false"/>
<objColor r="255" g="215" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0" y="7" z="0"/>
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="m">
<show object="true" label="false"/>
<objColor r="255" g="215" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-3.699864593426191" y="-2.5906730786722214" z="-3.699864593426191"/>
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="n">
<show object="true" label="false"/>
<objColor r="255" g="215" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="3.699864593426191" y="-4.409326921327779" z="29.59891674740953"/>
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="E" a1="D" a2="F"/>
<output a0="α"/>
</command>
<element type="angle" label="α">
<angleStyle val="1"/>
<value val="0.6981317007977319"/>
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="27" y="-13"/>
<labelMode val="2"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="45"/>
</element>
<command name="Angle">
<input a0="B" a1="A" a2="C"/>
<output a0="β"/>
</command>
<element type="angle" label="β">
<angleStyle val="1"/>
<value val="1.1344640137963142"/>
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="27" y="-13"/>
<labelMode val="2"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="45"/>
</element>
<command name="Angle">
<input a0="F" a1="E" a2="D"/>
<output a0="γ"/>
</command>
<element type="angle" label="γ">
<angleStyle val="1"/>
<value val="0.9599310885968814"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-55" y="-7"/>
<labelMode val="2"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="45"/>
</element>
<command name="Angle">
<input a0="C'" a1="B" a2="A"/>
<output a0="δ"/>
</command>
<element type="angle" label="δ">
<angleStyle val="1"/>
<value val="1.2217304763960306"/>
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-55" y="-7"/>
<labelMode val="2"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="45"/>
</element>
</construction>
</geogebra>