The Eyeball Theorem

Stage:4 and 5 Challenge Level:Challenge Level:3Challenge Level:3Challenge Level:3

In the interactive diagram below, the blue circle $C_1$ has a centre $X$ and fixed radius $R$, while the grey circle $C_2$ has a centre $Y$ and variable radius $r$. Two tangents to $C_2$ are drawn from $X$, cutting $C_1$ at points $A$ and $B$ and just touching $C_2$ at $P$ and $Q$. Tangents to $C_1$ are drawn from $Y$ cutting $C_2$ at points $C$ and $D$ and just touching $C_1$ at $R$ and $S$.

Click and drag the centres of the circles in the dynamic diagram below. As you change the distance between the centres of the circles and the radii what do you notice about the chords $AB$ and $CD$

Make and prove a conjecture about the chords $AB$ and $CD$.

Created with GeoGebra

NOTES AND BACKGROUND

This dynamic image is drawn using Geogebra, free software and very easy to use. You can download your own copy of Geogebra from http://www.geogebra.org/cms/ together with a good help manual and Quickstart for beginners. You may be surprised at how easy it is to draw the dynamic diagram above for yourself.

Doing mathematics often involves observing and explaining properties of `invariance', that is, what remains the same when the rest of the pattern changes according to certain rules that can be described in mathematical terms. NRICH dynamic mathematics problems allow you to alter the diagrams and change some properties, so that you can observe what remains invariant. This may lead you to a conjecture that you can prove. Proving the result in the case of The Eyeball Theorem uses only similar triangles.


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Published October 1998,September 2008.