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Three Similar Triangles: What Is It About?
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Copyright © 1996-2009 Alexander Bogomolny
Three Similar Triangles
The applet suggests a generalization of a statement about three isosceles triangles:
| Suppose A, B, C are arbitrary points on a straight line and X is a point not on the line. Construct similar and similarly oriented triangles ABX and BCY. If triangle XYZ is similar to triangles ABX and BCY but with a different orientation then Z is always collinear with A, B, and C! |
(This generalization and its simple synthetic proof have been suggested by Nathan Bowler.)
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Proof
First of all, by construction,
 XBY = 180o - ABX - CBY = XZY.
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The quadrilateral BXYZ is therefore cyclic. From here it follows that
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YBZ = YXZ = YBC,
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so that indeed B, C, and Z are collinear.
Nathan has also suggested a framework which underlies the origins of the above proof.
Copyright © 1996-2009 Alexander Bogomolny
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