Three Similar Triangles: What Is It About?
A Mathematical Droodle

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Explanation

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Copyright © 1996-2015 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 = 180° - ABX - CBY = XZY.

The quadrilateral BXYZ is therefore cyclic. From here it follows that

YBZ = YXZ = YBC,

so that indeed B, C, and Z are collinear.

Nathan has also suggested a framework which underlies the origins of the above proof.

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Copyright © 1996-2015 Alexander Bogomolny