Proportional Relationship Between Arcs and Inscribed Angles
2024-12-27 02:27
This GeoGebra applet dynamically demonstrates the proportional relationship between arc lengths and their corresponding inscribed angles in a circle. By constructing circles, chords, inscribed angles, and arcs, users can visually observe and verify the geometric theorem that the ratio of arcs equals the ratio of their subtended inscribed angles. It features real-time calculation and display of ang
In collections
Basic Properties of Circles
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.2.871.0" app="classic" platform="w" id="8f07b0b2-fd6e-492e-a2ca-df9874c72104" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="937" height="525" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.9978655282817502" 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 | 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="true" 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="935" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="true" location="3" size="200" tab="ALGEBRA" window="500,10,281,734" />
<view id="16" visible="false" inframe="false" stylebar="false" location="2" size="144" 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="4097" visible="false" inframe="false" stylebar="true" location="1" size="390" window="223,173,855,550" />
<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="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="20"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="934" height="522"/>
<coordSystem xZero="392.0976461706833" yZero="287.6749974126695" scale="36.92095533292915" yscale="36.92095533292915"/>
<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="0.1" distY="0.1" distTheta="0.5235987755982988"/>
</euclidianView>
<spreadsheetView>
<dimensions rows="100" columns="26"/></spreadsheetView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="0" max="0" step="0"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<expression label="r" exp="5"/>
<element type="numeric" label="r">
<value val="5"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="1"/>
<labelMode val="1"/>
<animation speed="1" type="0" playing="false"/>
</element>
<command name="Intersect">
<input a0="xAxis" a1="yAxis"/>
<output a0="O"/>
</command>
<element type="point" label="O">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelOffset x="-15" y="-10"/>
<labelMode val="0"/>
<pointSize val="1"/>
<pointStyle val="1"/>
<coords x="0" y="0" z="1"/>
</element>
<command name="Point">
<input a0="Sequence[Rotate[(r, 0), i°, O], i, 0, 359]"/>
<output a0="C"/>
</command>
<element type="point" label="C">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="1"/>
<labelOffset x="-3" y="39"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<coords x="3.8857298072848527" y="-3.146601955249189" z="1"/>
</element>
<command name="Sequence">
<input a0="Rotate[C, (2 * i°), O]" a1="i" a2="1" a3="179"/>
<output a0="l2"/>
</command>
<element type="list" label="l2">
<show object="false" label="false"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
</element>
<command name="Point">
<input a0="Sequence[Rotate[C, (2 * i°), O], i, 1, 179]"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<coords x="3.146601955249189" y="3.8857298072848527" z="1"/>
</element>
<command name="IndexOf">
<input a0="D" a1="l2"/>
<output a0="n2"/>
</command>
<element type="numeric" label="n2">
<value val="45"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<command name="Point">
<input a0="Sequence[Rotate[(r, 0), (i * 0.5°), O], i, 0, 719]"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="1"/>
<labelOffset x="-33" y="-3"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<coords x="-4.8063084796915945" y="-1.378186779084995" z="1"/>
</element>
<command name="Sequence">
<input a0="Rotate[A, (2 * i°), O]" a1="i" a2="1" a3="179"/>
<output a0="l1"/>
</command>
<element type="list" label="l1">
<show object="false" label="false"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
</element>
<command name="Point">
<input a0="l1"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<ggbscript onUpdate="
"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<coords x="3.5966990016932545" y="-3.4732918522949876" z="1"/>
</element>
<command name="IndexOf">
<input a0="B" a1="l1"/>
<output a0="n1"/>
</command>
<element type="numeric" label="n1">
<value val="60"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<command name="Point">
<input a0="l1 \ {B}"/>
<output a0="P"/>
</command>
<element type="point" label="P">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="1"/>
<labelOffset x="-18" y="2"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<ggbscript onUpdate="
"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<coords x="-4.045084971874737" y="2.938926261462365" z="1"/>
</element>
<expression label="A'" exp="A" type="point" />
<element type="point" label="A'">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-25" y="14"/>
<labelMode val="3"/>
<caption val="A"/>
<pointSize val="1"/>
<pointStyle val="0"/>
<coords x="-4.8063084796915945" y="-1.378186779084995" z="1"/>
</element>
<expression label="B'" exp="B" type="point" />
<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="2" y="27"/>
<labelMode val="3"/>
<caption val="B"/>
<pointSize val="1"/>
<pointStyle val="0"/>
<coords x="3.5966990016932545" y="-3.4732918522949876" z="1"/>
</element>
<expression label="C'" exp="C" type="point" />
<element type="point" label="C'">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="4" y="15"/>
<labelMode val="3"/>
<caption val="C"/>
<pointSize val="1"/>
<pointStyle val="0"/>
<coords x="3.8857298072848527" y="-3.146601955249189" z="1"/>
</element>
<expression label="D'" exp="D" type="point" />
<element type="point" label="D'">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="5" y="2"/>
<labelMode val="3"/>
<caption val="D"/>
<pointSize val="1"/>
<pointStyle val="0"/>
<coords x="3.146601955249189" y="3.8857298072848527" z="1"/>
</element>
<command name="Angle">
<input a0="A" a1="P" a2="B"/>
<output a0="α"/>
</command>
<element type="angle" label="α">
<angleStyle val="1"/>
<value val="1.0471975511965976"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="153"/>
<arcSize val="30"/>
<show object="true" label="true" ev="4"/>
<condition showObject="o1"/>
<objColor r="255" g="0" b="0" alpha="0.20000000298023224"/>
<layer val="1"/>
<labelOffset x="-3" y="36"/>
<labelMode val="2"/>
</element>
<command name="Point">
<input a0="l2 \ {D}"/>
<output a0="Q"/>
</command>
<element type="point" label="Q">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="1"/>
<labelOffset x="-16" y="5"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<coords x="-1.7918397477264993" y="4.6679021324860095" z="1"/>
</element>
<command name="Angle">
<input a0="C" a1="Q" a2="D"/>
<output a0="β"/>
</command>
<element type="angle" label="β">
<angleStyle val="1"/>
<value val="0.7853981633974482"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="153"/>
<arcSize val="30"/>
<show object="true" label="true" ev="4"/>
<condition showObject="o1"/>
<objColor r="0" g="0" b="255" alpha="0.20000000298023224"/>
<layer val="1"/>
<labelOffset x="21" y="30"/>
<labelMode val="2"/>
</element>
<command name="IsInRegion">
<input a0="P" a1="CircleArc[O, A, B]"/>
<output a0="OnArcAB"/>
</command>
<element type="boolean" label="OnArcAB">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<command name="If">
<input a0="¬OnArcAB" a1="CircleArc[O, A, B]" a2="CircleArc[O, B, A]"/>
<output a0="k"/>
</command>
<element type="conicpart" label="k">
<show object="true" label="true" ev="4"/>
<condition showObject="o2"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelOffset x="-66" y="27"/>
<labelMode val="2"/>
<lineStyle thickness="13" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="specific"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-25" A3="0" A4="0" A5="0"/>
</element>
<command name="IsInRegion">
<input a0="Q" a1="CircleArc[O, C, D]"/>
<output a0="OnArcCD"/>
</command>
<element type="boolean" label="OnArcCD">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<command name="If">
<input a0="¬OnArcCD" a1="CircleArc[O, C, D]" a2="CircleArc[O, D, C]"/>
<output a0="p"/>
</command>
<element type="conicpart" label="p">
<show object="true" label="true" ev="4"/>
<condition showObject="o2"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="1"/>
<labelOffset x="0" y="5"/>
<labelMode val="2"/>
<lineStyle thickness="8" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="specific"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-25" A3="0" A4="0" A5="0"/>
</element>
<element type="boolean" label="o1">
<value val="true"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="4" y="72"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="SetValue(ratio1, false)"/>
<caption val="Angles at the Centre"/>
</element>
<element type="boolean" label="o2">
<value val="true"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="4" y="112"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="SetValue(ratio2, false)"/>
<caption val="Arcs"/>
</element>
<command name="Circle">
<input a0="O" a1="r"/>
<output a0="c"/>
</command>
<element type="conic" label="c">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="specific"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-25" A3="0" A4="0" A5="0"/>
</element>
<expression label="text1" exp=""\text{拖拉各點，改變角的大小和位置。\\
圓周角的比和弧的比有甚麼關係？}""/>
<element type="text" label="text1">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<absoluteScreenLocation x="1" y="4"/>
</element>
<element type="boolean" label="labelang1">
<value val="true"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<element type="boolean" label="labelang2">
<value val="true"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<element type="boolean" label="labelarc1">
<value val="true"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<element type="boolean" label="labelarc2">
<value val="true"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<element type="boolean" label="labelchord1">
<value val="true"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<element type="boolean" label="labelchord2">
<value val="true"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<element type="boolean" label="ratios">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="738" y="71"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="SetValue(labelang1, true)
ShowLabel(α, labelang1)
SetValue(labelang2, true)
ShowLabel(β, labelang2)
SetValue(labelarc1, true)
ShowLabel(k, labelarc1)
SetValue(labelarc2, true)
ShowLabel(p, labelarc2)
SetValue(labelchord1, true)
ShowLabel(j, labelchord1)
SetValue(labelchord2, true)
ShowLabel(l, labelchord2)"/>
<caption val="Ratios"/>
</element>
<expression label="ratio1" exp=""\frac{∠APB}{∠CQD}=\frac{" + (LaTeX[α]) + "}{" + (LaTeX[β]) + "}=" + (LaTeX[FractionText[α / β]]) + """ />
<element type="text" label="ratio1">
<show object="false" label="true" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint x="8.281254668871593" y="3.0187453311284083" z="1"/>
</element>
<expression label="ratio2" exp=""\frac{\overset{\LARGE{\frown}}{\small{AB}}}{\overset{\LARGE{\frown}}{\small{CD}}}=\frac{" + (LaTeX[k]) + "}{" + (LaTeX[p]) + "}=" + (LaTeX[FractionText[k / p]]) + """ />
<element type="text" label="ratio2">
<show object="false" label="true" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint x="8.481254668871593" y="1.0187453311284083" z="1"/>
</element>
<element type="image" label="pic1">
<file name="46faf4dbc385f168a0cf0a9a5e23af7e/QSIP.png"/>
<inBackground val="false"/>
<absoluteScreenLocation x="6" y="515"/> <show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="1"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
</element>
<expression label="A''" exp="A" type="point" />
<element type="point" label="A''">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<pointSize val="4"/>
<pointStyle val="2"/>
<coords x="-4.8063084796915945" y="-1.378186779084995" z="1"/>
</element>
<expression label="P''" exp="P" type="point" />
<element type="point" label="P''">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<pointSize val="4"/>
<pointStyle val="2"/>
<coords x="-4.045084971874737" y="2.938926261462365" z="1"/>
</element>
<expression label="P'" exp="P" type="point" />
<element type="point" label="P'">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-18" y="-5"/>
<labelMode val="3"/>
<caption val="P"/>
<pointSize val="1"/>
<pointStyle val="0"/>
<coords x="-4.045084971874737" y="2.938926261462365" z="1"/>
</element>
<expression label="Q''" exp="Q" type="point" />
<element type="point" label="Q''">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<pointSize val="4"/>
<pointStyle val="2"/>
<coords x="-1.7918397477264993" y="4.6679021324860095" z="1"/>
</element>
<expression label="Q'" exp="Q" type="point" />
<element type="point" label="Q'">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-20" y="-6"/>
<labelMode val="3"/>
<caption val="Q"/>
<pointSize val="1"/>
<pointStyle val="0"/>
<coords x="-1.7918397477264993" y="4.6679021324860095" z="1"/>
</element>
<command name="Segment">
<input a0="P" a1="A"/>
<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"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<startStyle val="default"/>
<endStyle val="default"/>
<coords x="4.31711304054736" y="-0.7612235078168572" z="19.700268840168043"/>
</element>
<command name="Segment">
<input a0="P" a1="B"/>
<output a0="g"/>
</command>
<element type="segment" label="g">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<startStyle val="default"/>
<endStyle val="default"/>
<coords x="6.412218113757353" y="7.641783973567992" z="3.479327524001647"/>
</element>
<command name="Segment">
<input a0="Q" a1="C"/>
<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"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<startStyle val="default"/>
<endStyle val="default"/>
<coords x="7.814504087735198" y="5.6775695550113525" z="-12.499999999999996"/>
</element>
<command name="Segment">
<input a0="Q" a1="D"/>
<output a0="i"/>
</command>
<element type="segment" label="i">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<startStyle val="default"/>
<endStyle val="default"/>
<coords x="0.7821723252011568" y="4.938441702975688" z="-21.650635094610966"/>
</element>
<expression label="C''" exp="C" type="point" />
<element type="point" label="C''">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<pointSize val="4"/>
<pointStyle val="2"/>
<coords x="3.8857298072848527" y="-3.146601955249189" z="1"/>
</element>
<expression label="text12" exp=""" + (LaTeX[ratio1]) + "\\\\\\" + (LaTeX[ratio2]) + """ />
<element type="text" label="text12">
<show object="true" label="true" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint x="8.120681235321081" y="2.528912108887918" z="1"/>
</element>
<expression label="argP" exp="arg(P)" />
<element type="angle" label="argP">
<angleStyle val="0"/>
<value val="2.5132741228718345"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
</element>
<command name="IndexOf">
<input a0="B" a1="l1"/>
<output a0="n_{B}"/>
</command>
<element type="numeric" label="n_{B}">
<value val="60"/>
</element>
<command name="IndexOf">
<input a0="P" a1="l1"/>
<output a0="n_{P}"/>
</command>
<element type="numeric" label="n_{P}">
<value val="154"/>
</element>
</construction>
</geogebra>