Geometric Proof of the Tangent of Difference Formula
2020-04-10 01:54
This GeoGebra applet provides a visual and geometric proof of the tangent of difference formula. By constructing geometric objects such as circles, rays, orthogonal lines, and polygons, it creates a model containing two angles and their difference. Users can observe the proportional relationships of segment lengths to deeply understand the geometric meaning and derivation process of the trigonomet
In collections
Secondary Vocational Mathematics
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.574.0" app="classic" platform="w" id="8f63a212-7ba8-4f62-85a7-f0c565cd2632" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="613" height="589" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.3262642740619902" orientation="1" />
</panes>
<views>
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 51 | 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="1,1,1" size="500" window="100,100,600,400" />
<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="false" location="1,1" size="300" window="100,100,600,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="4097" visible="false" inframe="false" stylebar="true" location="1,1" size="395" window="100,100,700,550" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="614" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3" size="200" window="100,100,600,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<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="false" 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="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="43" visible="false" inframe="false" stylebar="false" location="1" size="450" 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="0"/>
<font size="12"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="613" height="588"/>
<coordSystem xZero="332.9938896702433" yZero="433.7295799309531" scale="260.0026947127056" yscale="260.0026947127058"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
<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"/>
</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="">
<command name="Intersect">
<input a0="xAxis" a1="yAxis"/>
<output a0="O_1"/>
</command>
<element type="point" label="O_1">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="-14" y="30"/>
<labelMode val="0"/>
<coords x="0" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="O_1" a1="0.5"/>
<output a0="c"/>
</command>
<element type="conic" label="c">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="20" typeHidden="1" opacity="178"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-0.25" A3="0" A4="0" A5="0"/>
<eqnStyle style="specific"/>
</element>
<expression label="text1" exp=""\tan(α-β)=\frac{\tanα-\tanβ}{1+
\tanα\tanβ}""/>
<element type="text" label="text1">
<show object="true" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint x="-0.639390201517425" y="-0.22146271277212914" z="1"/>
</element>
<command name="Intersect">
<input a0="c" a1="xAxis" a2="1"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="-13" y="23"/>
<labelMode val="0"/>
<coords x="-0.5" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<element type="point" label="B">
<show object="true" 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="0.7461696139116061" y="1.3604842839102587" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Ray">
<input a0="A" a1="B"/>
<output a0="f"/>
</command>
<element type="ray" label="f">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.3604842839102587" y="1.246169613911606" z="-0.6802421419551293"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="point" label="C">
<show object="true" 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="0.9038602872536792" y="0.5912614871196591" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Ray">
<input a0="A" a1="C"/>
<output a0="g"/>
</command>
<element type="ray" label="g">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.5912614871196591" y="1.4038602872536792" z="-0.29563074355982955"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Intersect">
<input a0="c" a1="xAxis" a2="2"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="8" y="13"/>
<labelMode val="0"/>
<coords x="0.5" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="OrthogonalLine">
<input a0="D" a1="xAxis"/>
<output a0="h"/>
</command>
<element type="line" label="h">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1" y="0" z="0.5"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="h" a1="f"/>
<output a0="E"/>
</command>
<element type="point" label="E">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="-3" y="-13"/>
<labelMode val="0"/>
<coords x="0.623084806955803" y="1.3604842839102587" z="1.246169613911606"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="h" a1="g"/>
<output a0="F"/>
</command>
<element type="point" label="F">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="7" y="26"/>
<labelMode val="0"/>
<coords x="0.7019301436268396" y="0.5912614871196591" z="1.4038602872536792"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="E" a1="A" a2="F"/>
<output a0="d"/>
</command>
<element type="conic" label="d">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="20" typeHidden="1" opacity="178"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-0.020098354988613232" A3="0" A4="0.22990164501138705" A5="-0.7564505798672326"/>
<eqnStyle style="specific"/>
</element>
<command name="Angle">
<input a0="O_1" a1="A" a2="B"/>
<output a0="α"/>
</command>
<element type="angle" label="α">
<angleStyle val="0"/>
<value val="0.8292250285871957"/>
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="22" y="-21"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="50"/>
</element>
<command name="Segment">
<input a0="A" a1="D"/>
<output a0="i"/>
</command>
<element type="segment" label="i">
<show object="true" label="true" ev="4"/>
<objColor r="75" g="0" b="130" alpha="0"/>
<layer val="0"/>
<labelOffset x="14" y="14"/>
<labelMode val="3"/>
<coords x="0" y="1" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="$\Huge 1$"/>
</element>
<command name="Center">
<input a0="d"/>
<output a0="O"/>
</command>
<element type="point" label="O">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="-22" y="31"/>
<labelMode val="0"/>
<coords x="-0.22990164501138705" y="0.7564505798672326" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="A" a1="B"/>
<output a0="l"/>
</command>
<element type="segment" label="l">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.3604842839102587" y="1.246169613911606" z="-0.6802421419551293"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="A" a1="C"/>
<output a0="m"/>
</command>
<element type="segment" label="m">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.5912614871196591" y="1.4038602872536792" z="-0.29563074355982955"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="E" a1="D"/>
<output a0="n"/>
</command>
<element type="segment" label="n">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="27" y="-8"/>
<labelMode val="3"/>
<coords x="1.0917328337350722" y="0" z="-0.5458664168675361"/>
<lineStyle thickness="13" type="0" typeHidden="1" opacity="89"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="$\tanα$"/>
</element>
<command name="Segment">
<input a0="D" a1="F"/>
<output a0="p"/>
</command>
<element type="segment" label="p">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-10" y="8"/>
<labelMode val="3"/>
<coords x="-0.421168325999393" y="0" z="0.2105841629996965"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="$tanβ$"/>
</element>
<command name="Angle">
<input a0="E" a1="D" a2="A"/>
<output a0="δ"/>
</command>
<element type="angle" label="δ">
<angleStyle val="0"/>
<value val="1.5707963267948966"/>
<show object="true" label="false" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
</element>
<command name="Segment">
<input a0="A" a1="E"/>
<output a0="b"/>
</command>
<element type="segment" label="b">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.0917328337350722" y="1" z="-0.5458664168675361"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="A" a1="F"/>
<output a0="g_1"/>
</command>
<element type="segment" label="g_1">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.421168325999393" y="1" z="-0.2105841629996965"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="D" a1="A" a2="C"/>
<output a0="β"/>
</command>
<element type="angle" label="β">
<angleStyle val="0"/>
<value val="0.39862071380182507"/>
<show object="true" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="42" y="2"/>
<labelMode val="3"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="191"/>
<arcSize val="70"/>
<caption val="$β$"/>
</element>
<command name="Angle">
<input a0="C" a1="A" a2="B"/>
<output a0="γ"/>
</command>
<element type="angle" label="γ">
<angleStyle val="0"/>
<value val="0.4306043147853708"/>
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="39" y="-26"/>
<labelMode val="3"/>
<lineStyle thickness="8" type="0" typeHidden="2" opacity="178"/>
<arcSize val="80"/>
<caption val="$α-β$"/>
</element>
<command name="Mirror">
<input a0="F" a1="O"/>
<output a0="F'"/>
</command>
<element type="point" label="F'">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="-25" y="-1"/>
<labelMode val="0"/>
<coords x="-1.347429722438398" y="1.5326403695715918" z="1.4038602872536792"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="F'" a1="E"/>
<output a0="j"/>
</command>
<element type="segment" label="j">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-32" y="16"/>
<labelMode val="3"/>
<coords x="0" y="1.4598032900227742" z="-1.5937151825123448"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="$1+\tanα\tanβ$"/>
</element>
<command name="Segment">
<input a0="F'" a1="F"/>
<output a0="k"/>
</command>
<element type="segment" label="k">
<show object="true" label="false" ev="4"/>
<objColor r="128" g="0" b="128" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.6705645077356792" y="1.4598032900227742" z="-0.9501051618151378"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="OrthogonalLine">
<input a0="A" a1="j"/>
<output a0="q"/>
</command>
<element type="line" label="q">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.4598032900227742" y="0" z="-0.7299016450113871"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="j" a1="q"/>
<output a0="G"/>
</command>
<element type="point" label="G">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.0655128227806578" y="2.326510666790767" z="2.1310256455613157"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Angle">
<input a0="F" a1="F'" a2="E"/>
<output a0="ε"/>
</command>
<element type="angle" label="ε">
<angleStyle val="0"/>
<value val="0.4306043147853707"/>
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="45" y="19"/>
<labelMode val="3"/>
<lineStyle thickness="8" type="0" typeHidden="2" opacity="178"/>
<arcSize val="80"/>
<caption val="$α-β$"/>
</element>
<command name="Segment">
<input a0="G" a1="A"/>
<output a0="r"/>
</command>
<element type="segment" label="r">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="20" y="0"/>
<labelMode val="3"/>
<coords x="1.0917328337350722" y="0" z="0.5458664168675361"/>
<lineStyle thickness="5" type="10" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="$\tanα$"/>
</element>
<command name="Segment">
<input a0="A" a1="F'"/>
<output a0="s"/>
</command>
<element type="segment" label="s">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="14" y="-26"/>
<labelMode val="0"/>
<coords x="-1.0917328337350722" y="-0.4598032900227741" z="-0.5458664168675361"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="F'" a1="G"/>
<output a0="t"/>
</command>
<element type="segment" label="t">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-24" y="-29"/>
<labelMode val="3"/>
<coords x="0" y="0.4598032900227741" z="-0.5019823487772724"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="$\tanα\tanβ$"/>
</element>
<command name="Angle">
<input a0="G" a1="A" a2="F'"/>
<output a0="ζ"/>
</command>
<element type="angle" label="ζ">
<angleStyle val="0"/>
<value val="0.39862071380182507"/>
<show object="true" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="-11" y="-37"/>
<labelMode val="3"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="191"/>
<arcSize val="50"/>
<caption val="$β$"/>
</element>
<command name="Segment">
<input a0="E" a1="F"/>
<output a0="a"/>
</command>
<element type="segment" label="a">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="22" y="-3"/>
<labelMode val="3"/>
<coords x="0.6705645077356792" y="0" z="-0.3352822538678396"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="$\tanα-\tanβ$"/>
</element>
<command name="Angle">
<input a0="F'" a1="E" a2="F"/>
<output a0="η"/>
</command>
<element type="angle" label="η">
<angleStyle val="0"/>
<value val="1.5707963267948966"/>
<show object="true" label="false" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
</element>
<command name="Polygon">
<input a0="F'" a1="F" a2="E"/>
<output a0="t1" a1="e" a2="f'" a3="f_1"/>
</command>
<element type="polygon" label="t1">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="e">
<show object="true" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0.6705645077356792" y="1.4598032900227742" z="-0.9501051618151378"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="f'">
<show object="true" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-0.6705645077356792" y="0" z="0.3352822538678396"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="f_1">
<show object="true" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0" y="-1.4598032900227742" z="1.5937151825123448"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="G" a1="E"/>
<output a0="h_1"/>
</command>
<element type="segment" label="h_1">
<show object="true" label="true" ev="4"/>
<objColor r="75" g="0" b="130" alpha="0"/>
<layer val="0"/>
<labelOffset x="-7" y="-29"/>
<labelMode val="3"/>
<coords x="0" y="1" z="-1.0917328337350722"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="$\Huge 1$"/>
</element>
</construction>
</geogebra>