Geometric Construction of Steiner Chain and Steiner Circles
2015-03-30 06:40
This GeoGebra applet demonstrates the geometric construction of a Steiner Chain. A Steiner chain is a closed sequence of circles, each tangent to two given non-intersecting circles and to its adjacent circles in the chain. Using GeoGebra's Sequence and geometric transformation commands, it dynamically visualizes the generation and tangency properties of these circles. This model helps intuitively
In collections
Geometric Transformations GeoGebra Book
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.42.0" id="87952a2b-7823-4252-9561-6c9b7aef047a" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="848" height="600" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.5459905660377359" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="784,564,700,489" />
<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="2" visible="false" inframe="false" stylebar="true" location="3,3" size="200" window="208,302,250,400" />
<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="false" location="3" size="463" window="100,100,600,400" />
<view id="16" visible="true" inframe="false" stylebar="false" location="1" size="365" 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" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<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 , 22 , 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" size="300" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 39 | 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 73 , 14 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6" position="1" help="true" />
<input show="true" cmd="true" top="false" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="12"/>
<graphicsSettings javaLatexFonts="false"/>
</gui>
<euclidianView>
<size width="463" height="471"/>
<coordSystem xZero="363.99765444785135" yZero="208.67903998509712" scale="77.520417540766" yscale="77.52041754076615"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" 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="10"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<euclidianView>
<viewNumber viewNo="2"/>
<size width="365" height="471"/>
<coordSystem xZero="361.36040505517684" yZero="255.92424698089033" scale="1655.677322160381" yscale="1655.6773221603858"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" 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="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"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0"/>
<coordStyle val="0"/>
<angleFromInvTrig val="false"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<euclidianView3D>
<coordSystem xZero="0.0" yZero="0.0" zZero="-1.5" scale="50.0" xAngle="20.0" zAngle="-60.0"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" gridType="0"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="2" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<plate show="true"/>
<bgColor r="255" g="255" b="255"/>
<clipping use="true" show="true" size="1"/>
<projection type="0"/>
</euclidianView3D>
<construction title="" author="" date="">
<element type="numeric" label="t">
<value val="-0.6"/>
<show object="true" label="false"/>
<objColor r="0" g="51" b="153" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="-2" max="2" absoluteScreenLocation="true" width="200.0" x="126.0" y="412.0" fixed="true" horizontal="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="n">
<value val="10.0"/>
<show object="false" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="1" max="20" absoluteScreenLocation="true" width="200.0" x="0.0" y="0.0" fixed="false" horizontal="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
</element>
<element type="point" label="B">
<show object="false" 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"/>
<selectionAllowed val="false"/>
<coords x="0.0" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="point" label="A">
<show object="false" 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"/>
<selectionAllowed val="false"/>
<coords x="-4.0" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="A" a1="B"/>
<output a0="a"/>
</command>
<element type="segment" label="a">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0.0" y="4.0" z="-0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Point">
<input a0="a"/>
<output a0="C"/>
</command>
<element type="point" label="C">
<show object="true" label="false"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-2.786332443762493" y="0.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Midpoint">
<input a0="C" a1="B"/>
<output a0="F"/>
</command>
<element type="point" label="F">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-1.3931662218812464" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="F" a1="B"/>
<output a0="e"/>
</command>
<element type="conic" label="e">
<show object="true" label="false"/>
<objColor r="255" g="102" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" 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="0.0" A3="0.0" A4="1.3931662218812464" A5="-0.0"/>
<eqnStyle style="specific"/>
</element>
<command name="Radius">
<input a0="e"/>
<output a0="r2"/>
</command>
<element type="numeric" label="r2">
<value val="1.3931662218812464"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<selectionAllowed val="false"/>
</element>
<command name="Midpoint">
<input a0="A" a1="C"/>
<output a0="E"/>
</command>
<element type="point" label="E">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-3.3931662218812466" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="E" a1="C"/>
<output a0="d"/>
</command>
<element type="conic" label="d">
<show object="false" label="false"/>
<objColor r="51" g="51" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" 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="11.145329775049971" A3="0.0" A4="3.3931662218812466" A5="-0.0"/>
<eqnStyle style="specific"/>
</element>
<command name="Radius">
<input a0="d"/>
<output a0="r1"/>
</command>
<element type="numeric" label="r1">
<value val="0.6068337781187538"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<selectionAllowed val="false"/>
</element>
<command name="Min">
<input a0="r1" a1="r2"/>
<output a0="r"/>
</command>
<element type="numeric" label="r">
<value val="0.6068337781187538"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<selectionAllowed val="false"/>
</element>
<command name="Circle">
<input a0="B" a1="r"/>
<output a0="f"/>
</command>
<element type="conic" label="f">
<show object="true" label="false"/>
<objColor r="255" g="51" b="102" alpha="0.2"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" 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="-0.36824723426588096" A3="0.0" A4="-0.0" A5="-0.0"/>
<eqnStyle style="specific"/>
</element>
<command name="Mirror">
<input a0="d" a1="f"/>
<output a0="d'"/>
</command>
<element type="conic" label="d'">
<show object="false" label="false" ev="3"/>
<objColor r="51" g="51" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" 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="0.012167071614878497" A3="0.0" A4="0.11211189815211864" A5="-0.0"/>
<eqnStyle style="specific"/>
</element>
<command name="Center">
<input a0="d'"/>
<output a0="G"/>
</command>
<element type="point" label="G">
<show object="false" label="true" ev="3"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-0.11211189815211864" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Mirror">
<input a0="A" a1="f"/>
<output a0="A'"/>
</command>
<element type="point" label="A'">
<show object="false" label="true" ev="3"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-0.09206180856647024" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Distance">
<input a0="G" a1="A'"/>
<output a0="r3'"/>
</command>
<element type="numeric" label="r3'">
<value val="0.020050089585648398"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<selectionAllowed val="false"/>
</element>
<command name="Rotate">
<input a0="A'" a1="90*°" a2="G"/>
<output a0="J"/>
</command>
<element type="point" label="J">
<show object="false" label="true" ev="3"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-0.11211189815211864" y="0.020050089585648398" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Midpoint">
<input a0="A" a1="B"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-2.0" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="D" a1="B"/>
<output a0="c"/>
</command>
<element type="conic" label="c">
<show object="true" label="false"/>
<objColor r="0" g="153" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" 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="0.0" A3="0.0" A4="2.0" A5="-0.0"/>
<eqnStyle style="specific"/>
</element>
<command name="Mirror">
<input a0="e" a1="f"/>
<output a0="e'"/>
</command>
<element type="conic" label="e'">
<show object="true" label="false" ev="3"/>
<objColor r="255" g="102" b="0" alpha="0.0" inverseFill="true"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<eigenvectors x0="0.0" y0="1.0" z0="1.0" x1="-1.0" y1="0.0" z1="1.0"/>
<matrix A0="0.0" A1="0.0" A2="-0.1841236171329405" A3="0.0" A4="-0.6965831109406232" A5="0.0"/>
<eqnStyle style="specific"/>
</element>
<command name="Mirror">
<input a0="c" a1="f"/>
<output a0="c'"/>
</command>
<element type="conic" label="c'">
<show object="true" label="false" ev="3"/>
<objColor r="0" g="153" b="0" alpha="0.0" inverseFill="true"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<eigenvectors x0="0.0" y0="1.0" z0="1.0" x1="-1.0" y1="0.0" z1="1.0"/>
<matrix A0="0.0" A1="0.0" A2="-0.18412361713294048" A3="0.0" A4="-1.0" A5="0.0"/>
<eqnStyle style="specific"/>
</element>
<command name="Mirror">
<input a0="C" a1="f"/>
<output a0="C'"/>
</command>
<element type="point" label="C'">
<show object="false" label="true" ev="3"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-0.13216198773776702" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Rotate">
<input a0="B" a1="90*°" a2="F"/>
<output a0="H"/>
</command>
<element type="point" label="H">
<show object="false" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-1.3931662218812464" y="1.3931662218812464" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Rotate">
<input a0="C" a1="90*°" a2="E"/>
<output a0="I"/>
</command>
<element type="point" label="I">
<show object="false" label="true"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-3.3931662218812466" y="0.6068337781187538" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Mirror">
<input a0="H" a1="f"/>
<output a0="H'"/>
</command>
<element type="point" label="H'">
<show object="false" label="true" ev="3"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-0.13216198773776702" y="0.13216198773776702" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Mirror">
<input a0="I" a1="f"/>
<output a0="I'"/>
</command>
<element type="point" label="I'">
<show object="false" label="true" ev="3"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-0.10516264598739032" y="0.01880728546982645" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Circle[G + ((2 * (J - G)) * (k + t)), r3']" a1="k" a2="(-n)" a3="n"/>
<output a0="listCircles"/>
</command>
<element type="list" label="listCircles">
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="255" alpha="0.24"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Zip">
<input a0="Mirror[c, f]" a1="c" a2="listCircles"/>
<output a0="list2"/>
</command>
<element type="list" label="list2">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.49"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Line">
<input a0="G" a1="J"/>
<output a0="b"/>
</command>
<element type="line" label="b">
<show object="false" label="false" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-0.020050089585648398" y="0.0" z="-0.0022478536015670676"/>
<lineStyle thickness="5" type="10" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Mirror">
<input a0="b" a1="f"/>
<output a0="b'"/>
</command>
<element type="conic" label="b'">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" 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="0.0" A3="0.0" A4="1.642320040671446" A5="-0.0"/>
<eqnStyle style="implicit"/>
</element>
</construction>
</geogebra>