Finding Radius of 3 Congruent Tangential Circles in Larger Circle

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SUMMARY

This discussion focuses on constructing three congruent tangential circles within a larger circle using the GeoGebra program on Ubuntu 11.4. The key formula derived for the radius of the smaller circles is SmallRadius = (LargeRadius * sin(60)) / (1 + sin(60)). The centers of the smaller circles form an equilateral triangle, providing a geometric foundation for the construction. Participants shared helpful hints and resources, including dynamic worksheets for visualizing the construction process.

PREREQUISITES
  • Understanding of basic geometric principles, particularly tangential circles.
  • Familiarity with trigonometric functions, specifically sine.
  • Experience using GeoGebra software for geometric constructions.
  • Knowledge of equilateral triangles and their properties.
NEXT STEPS
  • Explore advanced features of GeoGebra for dynamic geometry constructions.
  • Study the properties of tangential circles and their applications in geometry.
  • Learn about trigonometric identities and their use in geometric calculations.
  • Investigate the construction of multiple tangential circles in different configurations.
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Mathematicians, educators, students in geometry, and anyone interested in geometric constructions using software tools like GeoGebra.

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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