折纸中的角度关系探究
2021-01-19 17:00
本范例通过几何作图探究折纸过程中的角度关系。利用点、线段、多边形和镜像等工具,模拟纸张折叠的几何变换过程。通过构造折叠前后的对应点与线段,直观展示折痕与边缘所成角度的几何性质,帮助理解对称变换在角度计算中的应用,适合用于几何课程中的角度与对称性教学。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.620.0" app="classic" platform="d" id="2e74a44d-f399-4a14-876b-eb0a08ef4f4d" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1719" height="1200" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.745782431646306" 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 , 538 | 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="1" visible="true" inframe="false" stylebar="true" location="3" size="1682" window="100,100,600,400" />
<view id="2" visible="true" inframe="true" stylebar="true" location="3" size="400" tab="ALGEBRA" window="35,575,458,955" />
<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="4097" visible="true" inframe="true" stylebar="true" location="1" size="395" window="2256,512,446,906" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
</views>
<toolbar show="true" 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="true" cmd="true" top="algebra" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="2"/>
<font size="28"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1682" height="910"/>
<coordSystem xZero="764.6003952854725" yZero="509.01698383373827" scale="91.15395951621106" yscale="91.15395951621115"/>
<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>
<spreadsheetView>
<prefCellSize height="46"/>
</spreadsheetView>
<algebraView>
<mode val="1"/>
</algebraView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2.0" max="2.0" step="1.0"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="point" label="A">
<show object="false" label="false" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-5.0" y="-3.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="point" label="B">
<show object="false" label="false" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-5.0" y="1.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="point" label="C">
<show object="false" label="false" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="7.0" y="1.0000000000000002" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="point" label="D">
<show object="false" label="false" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="7.0" y="-3.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="PolyLine">
<input a0="A" a1="B" a2="C" a3="D"/>
<output a0="f"/>
</command>
<element type="polyline" label="f">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="8"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<command name="Point">
<input a0="f"/>
<output a0="E"/>
</command>
<element type="point" label="E">
<show object="true" label="false" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<coords x="-2.5736720218253426" y="1.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="A" a1="D"/>
<output a0="g"/>
</command>
<element type="segment" label="g">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="12.0" z="36.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Point">
<input a0="g"/>
<output a0="F"/>
</command>
<element type="point" label="F">
<show object="true" label="false" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<coords x="3.3394007932304075" y="-3.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="E" a1="F"/>
<output a0="h"/>
</command>
<element type="segment" label="h">
<show object="true" label="false" ev="4"/>
<condition showObject="i ≠ -5"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="4.0" y="5.9130728150557506" z="4.38161527224562"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Mirror">
<input a0="F" a1="h"/>
<output a0="F'"/>
</command>
<element type="point" label="F'">
<show object="true" label="false"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.339400793230408" y="-2.9999999999999996" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Mirror">
<input a0="E" a1="h"/>
<output a0="E'"/>
</command>
<element type="point" label="E'">
<show object="true" label="false" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2.5736720218253426" y="1.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="i" exp="x(E)" />
<element type="numeric" label="i">
<value val="-2.5736720218253426"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<command name="Polygon">
<input a0="A" a1="F" a2="E"/>
<output a0="t1" a1="e" a2="a" 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"/>
<condition showObject="i ≟ -5"/>
<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="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0.0" y="8.339400793230407" z="25.018202379691225"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="a">
<show object="false" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-4.0" y="-5.9130728150557506" z="-4.38161527224562"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="f_1">
<show object="false" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="4.0" y="-2.4263279781746574" z="12.721016065476029"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Mirror">
<input a0="A" a1="h"/>
<output a0="A'"/>
</command>
<element type="point" label="A'">
<show object="false" label="false" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.23621719648742068" y="4.740533389569301" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Mirror">
<input a0="F" a1="h"/>
<output a0="F'_{1}"/>
</command>
<element type="point" label="F'_{1}">
<show object="true" label="false"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.339400793230408" y="-2.9999999999999996" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Mirror">
<input a0="E" a1="h"/>
<output a0="E'_{1}"/>
</command>
<element type="point" label="E'_{1}">
<show object="true" label="false" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2.5736720218253426" y="1.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="A'" a1="F'_{1}" a2="E'_{1}"/>
<output a0="t1'" a1="e'_{1}" a2="a'" 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"/>
<condition showObject="i ≟ -5"/>
<objColor r="153" g="51" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="e'_{1}">
<show object="true" label="false" ev="4"/>
<condition showObject="i ≟ -5"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="7.7405333895693005" y="3.103183596742987" z="-16.539192550925222"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="a'">
<show object="false" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-3.9999999999999996" y="-5.9130728150557506" z="-4.381615272245619"/>
<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"/>
<condition showObject="i ≟ -5"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-3.7405333895693014" y="2.8098892183127635" z="-12.436795349750788"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Polygon">
<input a0="B" a1="E" a2="F" a3="A"/>
<output a0="q1" a1="b" a2="e_1" a3="f_2" a4="a_1"/>
</command>
<element type="polygon" label="q1">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<condition showObject="i ≠ -5"/>
<objColor r="153" g="51" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="b">
<show object="false" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0.0" y="2.4263279781746574" z="-2.4263279781746574"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="e_1">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="4.0" y="5.9130728150557506" z="4.38161527224562"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="f_2">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0.0" y="-8.339400793230407" z="-25.018202379691225"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="a_1">
<show object="true" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-4.0" y="0.0" z="-20.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Mirror">
<input a0="B" a1="h"/>
<output a0="B'"/>
</command>
<element type="point" label="B'">
<show object="false" label="false" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-3.4765355930668393" y="3.2520889923353766" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Mirror">
<input a0="E" a1="h"/>
<output a0="E'_{2}"/>
</command>
<element type="point" label="E'_{2}">
<show object="true" label="false" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2.5736720218253426" y="1.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Mirror">
<input a0="F" a1="h"/>
<output a0="F'_{2}"/>
</command>
<element type="point" label="F'_{2}">
<show object="true" label="false"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.339400793230408" y="-2.9999999999999996" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Mirror">
<input a0="A" a1="h"/>
<output a0="A'_{1}"/>
</command>
<element type="point" label="A'_{1}">
<show object="false" label="false" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.23621719648742068" y="4.740533389569301" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="B'" a1="E'_{2}" a2="F'_{2}" a3="A'_{1}"/>
<output a0="q1'" a1="b'" a2="e'_{2}" a3="f'_{2}" a4="a'_{1}"/>
</command>
<element type="polygon" label="q1'">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<condition showObject="i ≠ -5"/>
<objColor r="153" g="51" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="b'">
<show object="true" label="false" ev="4"/>
<condition showObject="i ≠ -5"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="2.2520889923353766" y="0.9028635712414967" z="4.89327485899289"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="e'_{2}">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="3.9999999999999996" y="5.9130728150557506" z="4.381615272245619"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="f'_{2}">
<show object="true" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-7.7405333895693005" y="-3.103183596742987" z="16.539192550925222"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="a'_{1}">
<show object="true" label="false" ev="4"/>
<condition showObject="i ≠ -5"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="1.4884443972339247" y="-3.7127527895542602" z="17.24883240345653"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Point">
<input a0="f"/>
<output a0="G"/>
</command>
<element type="point" label="G">
<show object="true" label="false" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
<coords x="7.0" y="-1.7879971098488185" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="j" exp="x(G)" />
<element type="numeric" label="j">
<value val="7.0"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<command name="Polygon">
<input a0="F" a1="D" a2="G"/>
<output a0="t2" a1="g_1" a2="f_3" a3="d"/>
</command>
<element type="polygon" label="t2">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<condition showObject="j ≟ 7"/>
<objColor r="153" g="51" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="g_1">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0.0" y="3.6605992067695925" z="10.981797620308777"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="f_3">
<show object="true" label="false" ev="4"/>
<condition showObject="j ≟ 7"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-1.2120028901511815" y="0.0" z="8.48402023105827"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="d">
<show object="true" label="false" ev="4"/>
<condition showObject="j ≟ 7"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="1.2120028901511815" y="-3.6605992067695925" z="-15.02916103307718"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Mirror">
<input a0="F" a1="d"/>
<output a0="F'_{3}"/>
</command>
<element type="point" label="F'_{3}">
<show object="true" label="false"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.339400793230408" y="-3.000000000000001" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Mirror">
<input a0="D" a1="d"/>
<output a0="D'"/>
</command>
<element type="point" label="D'">
<show object="false" label="false" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="6.276714847274175" y="-0.8154696842297806" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Mirror">
<input a0="G" a1="d"/>
<output a0="G'"/>
</command>
<element type="point" label="G'">
<show object="true" label="false"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="7.0" y="-1.78799710984882" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="F'_{3}" a1="D'" a2="G'"/>
<output a0="t2'" a1="g'" a2="f'_{3}" a3="d'"/>
</command>
<element type="polygon" label="t2'">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<condition showObject="j ≟ 7"/>
<objColor r="153" g="51" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="g'">
<show object="true" label="false" ev="4"/>
<condition showObject="j ≟ 7"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-2.1845303157702203" y="2.9373140540437674" z="16.10696443145025"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="f'_{3}">
<show object="true" label="false" ev="4"/>
<condition showObject="j ≟ 7"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0.9725274256190395" y="0.7232851527258246" z="-5.51446021666294"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="d'">
<show object="true" label="false" ev="4"/>
<condition showObject="j ≟ 7"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="1.2120028901511808" y="-3.660599206769592" z="-15.02916103307718"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Polygon">
<input a0="F" a1="D" a2="C" a3="G"/>
<output a0="q2" a1="f_4" a2="d_1" a3="c" a4="g_2"/>
</command>
<element type="polygon" label="q2">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<condition showObject="j ≠ 7"/>
<objColor r="153" g="51" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="f_4">
<show object="true" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0.0" y="3.6605992067695925" z="10.981797620308777"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="d_1">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-4.0" y="0.0" z="28.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="c">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="2.7879971098488188" y="0.0" z="-19.515979768941733"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="g_2">
<show object="true" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="1.2120028901511815" y="-3.6605992067695925" z="-15.02916103307718"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Mirror">
<input a0="F" a1="g_2"/>
<output a0="F'_{4}"/>
</command>
<element type="point" label="F'_{4}">
<show object="true" label="false"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.339400793230408" y="-3.000000000000001" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Mirror">
<input a0="D" a1="g_2"/>
<output a0="D'_{1}"/>
</command>
<element type="point" label="D'_{1}">
<show object="false" label="false" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="6.276714847274175" y="-0.8154696842297806" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Mirror">
<input a0="C" a1="g_2"/>
<output a0="C'"/>
</command>
<element type="point" label="C'">
<show object="false" label="false" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="8.663788866992412" y="-4.025123501136866" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Mirror">
<input a0="G" a1="g_2"/>
<output a0="G'_{1}"/>
</command>
<element type="point" label="G'_{1}">
<show object="true" label="false"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="7.0" y="-1.78799710984882" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="F'_{4}" a1="D'_{1}" a2="C'" a3="G'_{1}"/>
<output a0="q2'" a1="f'_{4}" a2="d'_{1}" a3="c'" a4="g'_{1}"/>
</command>
<element type="polygon" label="q2'">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<condition showObject="j ≠ 7"/>
<objColor r="153" g="51" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="f'_{4}">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-2.1845303157702203" y="2.9373140540437674" z="16.10696443145025"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="d'_{1}">
<show object="true" label="false" ev="4"/>
<condition showObject="j ≠ 7"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="3.2096538169070854" y="2.387074019718237" z="-18.19949527009819"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="c'">
<show object="true" label="false" ev="4"/>
<condition showObject="j ≠ 7"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-2.237126391288046" y="-1.6637888669924124" z="12.685035053435243"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="g'_{1}">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="1.2120028901511808" y="-3.660599206769592" z="-15.02916103307718"/>
<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="k"/>
</command>
<element type="segment" label="k">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="8.339400793230407" z="25.018202379691225"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="D" a1="F"/>
<output a0="l"/>
</command>
<element type="segment" label="l">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="-3.6605992067695925" z="-10.981797620308777"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="G" a1="F" a2="E"/>
<output a0="α"/>
</command>
<element type="angle" label="α">
<angleStyle val="0"/>
<value val="2.227101784012387"/>
<show object="true" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="-3" y="55"/>
<labelMode val="2"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="50"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
</element>
<command name="Angle">
<input a0="g'" a1="f'_{2}"/>
<output a0="β"/>
</command>
<element type="angle" label="β">
<angleStyle val="0"/>
<value val="1.3126109144349805"/>
<show object="true" label="true" ev="4"/>
<objColor r="255" g="51" b="102" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="24" y="-70"/>
<labelMode val="2"/>
<lineStyle thickness="7" type="0" typeHidden="2" opacity="178"/>
<arcSize val="65"/>
<ggbscript val=""/>
<ggbscript onUpdate=""/>
</element>
</construction>
</geogebra>