Constant Parallelogram Area Property of Hyperbola
2019-06-04 11:01
This GeoGebra example demonstrates a fundamental geometric property of hyperbolas: for any point P on the hyperbola, the parallelogram ORPQ formed by drawing lines through P parallel to the asymptotes has a constant area equal to ab/2, where a and b are the semi-major and semi-minor axes respectively. The applet allows toggling the display of foci, vertices, asymptotes, and the auxiliary rectangle
In collections
Conic Sections
Loading the math board and drawing, please wait…
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.541.0" app="classic" platform="w" id="4e8472a4-af2a-4533-9dda-d237a8f837ae" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="970" height="654" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="1" 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 || 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 || 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="970" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="true" location="3" size="298" window="100,100,600,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="300" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="300" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="false" stylebar="false" location="1" size="480" 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="437" window="795,303,700,550" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="900" 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 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="16"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="970" height="654"/>
<coordSystem xZero="484.35629426187995" yZero="324.0877339368999" scale="58.703730484818934" yscale="58.7037270710333"/>
<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" tickDistance="1" tickExpression="1"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="1" tickExpression="1"/>
</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="">
<expression label="O" exp="(0, 0)" type="point"/>
<element type="point" label="O">
<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="0" y="0" z="1"/>
<pointSize val="9"/>
<pointStyle val="0"/>
</element>
<expression label="i" exp="(1, 0)" type="vector"/>
<element type="vector" label="i">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="1" y="0" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
</element>
<expression label="j" exp="(0, 1)" type="vector"/>
<element type="vector" label="j">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="0" y="1" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
</element>
<command name="Point">
<input a0="Ray[O, O + i]"/>
<output a0="F"/>
</command>
<element type="point" label="F">
<show object="true" label="false" ev="4"/>
<condition showObject="showFocus ≠ 0"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="5" y="0" z="1"/>
<pointSize val="7"/>
<pointStyle val="1"/>
</element>
<command name="Point">
<input a0="Segment[O, F]"/>
<output a0="A_1"/>
</command>
<element type="point" label="A_1">
<show object="true" label="false" ev="4"/>
<condition showObject="showVertex ≠ 0"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="4" y="0" z="1"/>
<pointSize val="9"/>
<pointStyle val="0"/>
</element>
<expression label="c" exp="x(F)" />
<element type="numeric" label="c">
<value val="5"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<expression label="a" exp="x(A_1)" />
<element type="numeric" label="a">
<value val="4"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<expression label="b" exp="sqrt(c^(2) - a^(2))" />
<element type="numeric" label="b">
<value val="3"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<command name="Circle">
<input a0="O" a1="F"/>
<output a0="d"/>
</command>
<element type="conic" label="d">
<show object="true" label="false" ev="4"/>
<condition showObject="showBox ≠ 0"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="3" 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="-25" A3="0" A4="0" A5="0"/>
<eqnStyle style="specific"/>
</element>
<expression label="B" exp="A_1 + (b * j)" type="point" />
<element type="point" label="B">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="4" y="3" z="1"/>
<pointSize val="9"/>
<pointStyle val="0"/>
</element>
<expression label="C" exp="A_1 - (b * j)" type="point" />
<element type="point" label="C">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="4" y="-3" z="1"/>
<pointSize val="9"/>
<pointStyle val="0"/>
</element>
<expression label="D" exp="(-B)" type="point" />
<element type="point" label="D">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-4" y="-3" z="1"/>
<pointSize val="9"/>
<pointStyle val="0"/>
</element>
<expression label="E" exp="(-C)" type="point" />
<element type="point" label="E">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-4" y="3" z="1"/>
<pointSize val="9"/>
<pointStyle val="0"/>
</element>
<command name="Line">
<input a0="B" a1="D"/>
<output a0="L1"/>
</command>
<element type="line" label="L1">
<show object="true" label="false" ev="4"/>
<condition showObject="0 < a < c ∧ showAsympotote"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="6" y="-8" z="0"/>
<lineStyle thickness="5" type="10" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Line">
<input a0="C" a1="E"/>
<output a0="L2"/>
</command>
<element type="line" label="L2">
<show object="true" label="false" ev="4"/>
<condition showObject="0 < a < c ∧ showAsympotote"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-6" y="-8" z="0"/>
<lineStyle thickness="5" type="10" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Polygon">
<input a0="{B, C, D, E}"/>
<output a0="box"/>
</command>
<element type="polygon" label="box">
<lineStyle thickness="3" type="20" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<condition showObject="showBox ≠ 0"/>
<objColor r="51" g="51" b="255" alpha="0.25"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<expression label="e" exp="x^(2) / a^(2) - y^(2) / b^(2) = 1" type="conic" />
<element type="conic" label="e">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<lineStyle thickness="9" type="0" typeHidden="1" opacity="178"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="-1.7777777777777777" A2="-16" A3="0" A4="0" A5="0"/>
<eqnStyle style="implicit"/>
</element>
<expression label="h" exp="x = 0" type="line"/>
<element type="line" label="h">
<show object="true" label="false" ev="4"/>
<condition showObject="showBox ≠ 0"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<coords x="1" y="0" z="0"/>
<lineStyle thickness="3" type="20" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<expression label="k" exp="y = 0" type="line"/>
<element type="line" label="k">
<show object="true" label="false" ev="4"/>
<condition showObject="showBox ≠ 0"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<coords x="0" y="1" z="0"/>
<lineStyle thickness="3" type="20" typeHidden="1" opacity="178"/>
<eqnStyle style="explicit"/>
</element>
<expression label="rays" exp="{Ray[F, F + i], Ray[(-F), (-F) - i]}" />
<element type="list" label="rays">
<show object="true" label="true" ev="4"/>
<condition showObject="a ≟ c"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="9" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<expression label="l" exp="x = 0" type="line"/>
<element type="line" label="l">
<show object="true" label="false" ev="4"/>
<condition showObject="a ≟ 0"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<coords x="1" y="0" z="0"/>
<lineStyle thickness="9" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<expression label="A_2" exp="(-A_1)" type="point" />
<element type="point" label="A_2">
<show object="true" label="false" ev="4"/>
<condition showObject="showVertex ≠ 0"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-4" y="0" z="1"/>
<pointSize val="9"/>
<pointStyle val="0"/>
</element>
<expression label="F_2" exp="(-F)" type="point" />
<element type="point" label="F_2">
<show object="true" label="false" ev="4"/>
<condition showObject="showFocus ≠ 0"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-5" y="0" z="1"/>
<pointSize val="7"/>
<pointStyle val="1"/>
</element>
<element type="boolean" label="showFocus">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="5" y="35"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val="焦點"/>
</element>
<element type="boolean" label="showVertex">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="5" y="5"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val="頂點"/>
</element>
<element type="boolean" label="showAsympotote">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="5" y="65"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val="漸近線"/>
</element>
<element type="boolean" label="showBox">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="5" y="95"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val="a, b, c"/>
</element>
<command name="Point">
<input a0="e"/>
<output a0="P"/>
</command>
<element type="point" label="P">
<show object="true" label="false" ev="4"/>
<condition showObject="showP ∧ (¬showBox)"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="4.1761297017527" y="0.9000185266409055" z="1"/>
<pointSize val="9"/>
<pointStyle val="0"/>
</element>
<expression label="showP" exp="true"/>
<element type="boolean" label="showP">
<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>
<command name="Intersect">
<input a0="L2" a1="Line[P, L1]"/>
<output a0="R"/>
</command>
<element type="point" label="R">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="142.85303997911166" y="-107.13977998433374" z="96"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="L1" a1="Line[P, L2]"/>
<output a0="Q"/>
</command>
<element type="point" label="Q">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="258.05541138914754" y="193.54155854186064" z="96"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="{O, R, P, Q}"/>
<output a0="ORPQ"/>
</command>
<element type="polygon" label="ORPQ">
<lineStyle thickness="3" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<condition showObject="¬showBox"/>
<objColor r="102" g="204" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="Distance">
<input a0="P" a1="Q"/>
<output a0="d1"/>
</command>
<element type="numeric" label="d1">
<value val="1.860065624728016"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<command name="Distance">
<input a0="P" a1="R"/>
<output a0="d2"/>
</command>
<element type="numeric" label="d2">
<value val="3.3600965024628584"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<expression label="d1d2" exp="(d1 * d2)" />
<element type="numeric" label="d1d2">
<value val="6.249999999999998"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<command name="Polygon">
<input a0="{O, A_1, B}"/>
<output a0="t1"/>
</command>
<element type="polygon" label="t1">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<condition showObject="showBox ≠ 0"/>
<objColor r="102" g="204" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<expression label="text1" exp=""\theta""/>
<element type="text" label="text1">
<show object="true" label="false" ev="40"/>
<condition showObject="showBox ≠ 0"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="7" style="1"/>
<startPoint x="0.949136304110026" y="0.7269389344702024" z="1"/>
</element>
<command name="Angle">
<input a0="A_1" a1="O" a2="B"/>
<output a0="α"/>
</command>
<element type="angle" label="α">
<angleStyle val="0"/>
<value val="0.6435011087932844"/>
<show object="true" label="false" ev="4"/>
<condition showObject="showBox ≠ 0"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<decoration type="1"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="40"/>
<selectionAllowed val="false"/>
</element>
<expression label="text3" exp=""b""/>
<element type="text" label="text3">
<show object="true" label="false" ev="40"/>
<condition showObject="showBox ≠ 0"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="7" style="1"/>
<startPoint exp="Midpoint[A_1, B]"/>
<labelOffset x="-2" y="-7"/>
</element>
<expression label="text4" exp=""c""/>
<element type="text" label="text4">
<show object="true" label="false" ev="40"/>
<condition showObject="showBox ≠ 0"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="7" style="1"/>
<startPoint exp="Midpoint[O, B]"/>
<labelOffset x="-15" y="-37"/>
</element>
<expression label="text2" exp=""a""/>
<element type="text" label="text2">
<show object="true" label="false" ev="40"/>
<condition showObject="showBox ≠ 0"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="7" style="1"/>
<startPoint exp="Midpoint[O, A_1]"/>
<labelOffset x="6" y="0"/>
</element>
<expression label="text5" exp=""▱=\frac{ab}{2}""/>
<element type="text" label="text5">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2" size="16" style="1"/>
<absoluteScreenLocation x="404" y="36"/>
</element>
<element type="boolean" label="showProof">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="5" y="125"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val="證明"/>
</element>
<expression label="text6" exp=""▱=d_1 d_2 \sin(2\theta)""/>
<element type="text" label="text6">
<show object="true" label="true" ev="40"/>
<condition showObject="showProof ≠ 0"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="3"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2" size="16" style="1"/>
<absoluteScreenLocation x="74" y="196"/>
</element>
<expression label="text7" exp=""=\frac{c^2}{4} \cdot 2\sin\theta\cos\theta""/>
<element type="text" label="text7">
<show object="true" label="false" ev="40"/>
<condition showObject="showProof ≠ 0"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2" size="16" style="1"/>
<absoluteScreenLocation x="120" y="249"/>
</element>
<expression label="text8" exp=""=\frac{c^2}{4} \cdot 2 \cdot\frac{b}{c}\cdot\frac{a}{c}""/>
<element type="text" label="text8">
<show object="true" label="false" ev="40"/>
<condition showObject="showProof ≠ 0"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2" size="16" style="1"/>
<absoluteScreenLocation x="123" y="347"/>
</element>
<expression label="text9" exp=""=\frac{ab}{2}""/>
<element type="text" label="text9">
<show object="true" label="false" ev="40"/>
<condition showObject="showProof ≠ 0"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2" size="16" style="1"/>
<absoluteScreenLocation x="124" y="455"/>
</element>
<expression label="text10" exp=""d_1""/>
<element type="text" label="text10">
<show object="true" label="false" ev="40"/>
<condition showObject="showProof ≠ 0"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="1"/>
<startPoint exp="Midpoint[P, Q]"/>
<labelOffset x="-6" y="-22"/>
</element>
<expression label="text11" exp=""d_2""/>
<element type="text" label="text11">
<show object="true" label="false" ev="40"/>
<condition showObject="showProof ≠ 0"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="1"/>
<startPoint exp="Midpoint[P, R]"/>
<labelOffset x="-6" y="0"/>
</element>
<command name="Angle">
<input a0="Q" a1="P" a2="R"/>
<output a0="β"/>
</command>
<element type="angle" label="β">
<angleStyle val="1"/>
<emphasizeRightAngle val="false"/>
<value val="1.287002217586569"/>
<show object="true" label="false" ev="4"/>
<condition showObject="showProof ≠ 0"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="-7" y="80"/>
<labelMode val="1"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="20"/>
<selectionAllowed val="false"/>
</element>
<expression label="text12" exp=""2\theta""/>
<element type="text" label="text12">
<show object="true" label="false" ev="40"/>
<condition showObject="showProof ≠ 0"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="1"/>
<startPoint exp="P"/>
<labelOffset x="-53" y="-12"/>
</element>
</construction>
</geogebra>