Geometric Construction of Triangle Incenter and Angle Bisectors
2013-11-19 13:02
This applet demonstrates the geometric construction of a triangle's incenter and incircle. By drawing the angle bisectors, their intersection is the incenter. An orthogonal line is dropped to a side, and the incircle is drawn using this perpendicular distance as the radius. It visually illustrates that angle bisectors are concurrent and the incenter is equidistant from all three sides.
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="4.0" id="a3238e44-99ba-4cd7-a8a2-8f2988c47a78" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="875" height="713" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.26857142857142857" orientation="1" />
</panes>
<views>
<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="false" location="1,1" size="300" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="false" location="1" size="873" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3" size="235" window="100,100,250,400" />
<view id="16" visible="false" inframe="true" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="32" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="64" visible="false" inframe="true" stylebar="false" location="1" size="150" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 39 59 || 1 501 67 , 5 19 , 72 | 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 | 30 29 54 32 31 33 | 17 26 62 , 14 66 68 | 25 52 60 61 || 40 41 42 , 27 28 35 , 6" />
<input show="true" cmd="true" top="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="12"/>
<graphicsSettings javaLatexFonts="false"/>
<consProtColumns col0="true" col1="true" col2="false" col3="true" col4="false" col5="true" col6="true" col7="false"/>
<consProtocol useColors="true" addIcons="false" showOnlyBreakpoints="false"/>
</gui>
<euclidianView>
<size width="873" height="553"/>
<coordSystem xZero="215.00000000000003" yZero="315.00000000000006" scale="50.00000000000001" yscale="50.00000000000001"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="13" gridType="0"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="204" g="204" b="204"/>
<lineStyle axes="1" grid="10"/>
<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"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0"/>
<coordStyle val="0"/>
<angleFromInvTrig val="false"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="point" label="A">
<show object="true" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="4.36" y="3.86" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="point" label="B">
<show object="true" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="1.62" y="2.14" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="point" label="C">
<show object="true" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="0.94" y="-2.38" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="point" label="D">
<show object="true" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="6.76" y="-0.36" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="A" a1="B" a2="C" a3="D"/>
<output a0="poly1" a1="a" a2="b" a3="c" a4="d"/>
</command>
<element type="polygon" label="poly1">
<lineStyle thickness="4" type="0" typeHidden="1"/>
<show object="true" label="false"/>
<objColor r="255" g="0" b="102" alpha="0.1"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="a">
<show object="true" label="false"/>
<objColor r="255" g="0" b="102" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.7199999999999998" y="-2.74" z="3.0772000000000004"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="b">
<show object="true" label="false"/>
<objColor r="255" g="0" b="102" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="4.52" y="-0.6800000000000002" z="-5.8672"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="c">
<show object="true" label="false"/>
<objColor r="255" g="0" b="102" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2.02" y="5.82" z="15.750399999999999"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="d">
<show object="true" label="false"/>
<objColor r="255" g="0" b="102" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-4.22" y="-2.3999999999999995" z="27.6632"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="AngularBisector">
<input a0="B" a1="A" a2="D"/>
<output a0="e"/>
</command>
<element type="line" label="e">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.9697566685154193" y="-0.24407376727103494" z="-3.2860143330610336"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="AngularBisector">
<input a0="A" a1="D" a2="C"/>
<output a0="f"/>
</command>
<element type="line" label="f">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.35209822925816997" y="-0.9359630531988222" z="2.043237330633653"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="AngularBisector">
<input a0="D" a1="C" a2="B"/>
<output a0="g"/>
</command>
<element type="line" label="g">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.7693173706319698" y="0.6388667961663935" z="2.243661303270068"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="e" a1="f"/>
<output a0="E"/>
</command>
<element type="point" label="E">
<show object="true" label="true"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.5742886407434655" y="0.8244431987538732" z="0.9935943535881121"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="e" a1="f"/>
<output a0="F"/>
</command>
<element type="point" label="F">
<show object="true" label="true"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.5742886407434655" y="0.8244431987538732" z="0.9935943535881121"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="OrthogonalLine">
<input a0="E" a1="a"/>
<output a0="h"/>
</command>
<element type="line" label="h">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.7224485288314275" y="1.7089822881715526" z="-11.211593177493759"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="h" a1="a"/>
<output a0="G"/>
</command>
<element type="point" label="G">
<show object="true" label="true"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="25.4608850091714" y="27.661458878209334" z="10.398958504653182"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="h" a1="a"/>
<output a0="H"/>
</command>
<element type="point" label="H">
<show object="true" label="true"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="25.4608850091714" y="27.661458878209334" z="10.398958504653182"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="E" a1="G"/>
<output a0="k"/>
</command>
<element type="conic" label="k">
<show object="true" label="false"/>
<objColor r="255" g="0" b="102" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="4" type="10" typeHidden="1"/>
<eigenvectors x0="1.0" y0="0.0" z0="1.0" x1="-0.0" y1="1.0" z1="1.0"/>
<matrix A0="1.0" A1="1.0" A2="8.959402402963827" A3="0.0" A4="-3.597331876772282" A5="-0.8297583372697391"/>
<eqnStyle style="specific"/>
</element>
</construction>
</geogebra>