Finding Radius of 3 Congruent Tangential Circles in Larger Circle

BobFijiwinkle
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Hello everyone!

I'm trying to find out how to precisely construct three congruent circles inside a larger circle, each tangential to both the outer circle and the other two circles. For example:
[PLAIN]http://img4.imageshack.us/img4/1044/verybasicdrawing.png

An image I found on the internet, but with six:
[URL]http://etc.usf.edu/clipart/32400/32425/_32425_lg.gif[/URL]

I'm using the Geogebra program (http://www.geogebra.org/" ) on Ubuntu 11.4 Natty, but I'm mainly looking for the geometry behind it.

I saw on https://www.physicsforums.com/showpost.php?p=1822287&postcount=2" (thanks ZharAngel) how to find a point at a given angle and distance from another point. However, I need to find the distance of the inner circles' center points from the outside circle's center point.

Help anyone?

Thanks!
BF
 
Last edited by a moderator:
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Hello BF! :smile:

Hint: if the smaller circles have radius r, then their centres form an equilateral triangle of length 2r …

how far is its centre from each vertex? :wink:​
 
Excellent, thank you! Good hint.

The formula I have (that seems to work) is

SmallRadius = \frac{LargeRadius \sin 60}{1 + \sin 60}

If you paste the following into an html document, you can see it in action.

Code: 
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<title>Congruent Tangential Circles - GeoGebra Dynamic worksheet</title>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<meta name="generator" content="GeoGebra" />
<style type="text/css"><!--body { font-family:Arial,Helvetica,sans-serif; margin-left:40px }--></style>
</head>
<body>
<table border="0" width="1280">
<tr><td>
<h2>Congruent Tangential Circles</h2>
<p>
This construction shows three tangential circles inside a larger circle. The inside circles are tangential to both the two other circles and the outer circle.</p>


<applet name="ggbApplet" code="geogebra.GeoGebraApplet" archive="geogebra.jar"
	codebase="[PLAIN]http://www.geogebra.org/webstart/3.2/unsigned/"[/PLAIN] 
	width="1280" height="838" mayscript="true">
	<param name="ggbBase64" value="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"/>
	<param name="image" value="[PLAIN]http://www.geogebra.org/webstart/loading.gif"[/PLAIN]   />
	<param name="boxborder" value="false"  />
	<param name="centerimage" value="true"  />
	<param name="java_arguments" value="-Xmx512m" />
	<param name="cache_archive" value="geogebra.jar, geogebra_main.jar, geogebra_gui.jar, geogebra_cas.jar, geogebra_export.jar, geogebra_properties.jar" />
	<param name="cache_version" value="3.2.44.0, 3.2.44.0, 3.2.44.0, 3.2.44.0, 3.2.44.0, 3.2.44.0" />
	<param name="framePossible" value="false" />
	<param name="showResetIcon" value="true" />
	<param name="showAnimationButton" value="true" />
	<param name="enableRightClick" value="false" />
	<param name="errorDialogsActive" value="true" />
	<param name="enableLabelDrags" value="false" />
	<param name="showMenuBar" value="false" />
	<param name="showToolBar" value="false" />
	<param name="showToolBarHelp" value="false" />
	<param name="showAlgebraInput" value="false" />
	<param name="allowRescaling" value="true" />
Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (<a href="http://java.sun.com/getjava">Click here to install Java now</a>)
</applet>

<p>
Use the Radius slider to adjust the large circle radius, and the Angle slider to adjust the position of the smaller circles inside the larger one, as well as their colors.</p>
<p><span style="font-size:small">Bob Fijiwinkle, Created with <a href="[PLAIN]http://www.geogebra.org/"[/PLAIN]  target="_blank" >GeoGebra</a></span></p></td></tr>
</table></body>
</html>

Thanks very much, tiny-tim!

BF
 
Last edited by a moderator:

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