### Three Similar Triangles II: What Is It About?

A Mathematical Droodle

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. On line BX choose a point X', on line BY choose a point Y'. If triangle X'Y'Z' is similar to triangles ABX and BCY but with a different orientation then Z' is always collinear with A, B, and C!

This beautiful result and its proof are due to Nathan Bowler. This is a clear generalization of the case where

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Copyright © 1996-2018 Alexander Bogomolny

### Three Similar Triangles

Let the origin O coincide with B. Define a function F so that for any P and Q, F(P, Q) is the unique point R with PQR similar to ABX but with a different orientation. (So that, e.g., Z = F(X, Y).) F is clearly a linear function. Now define a function G of pairs of scalars

In particular, the image of G is either the whole plane or a straight line through O (it is not the point O since X and Y are nonzero). Now let k be such that

Now, let R = G(1, 0). Since

In particular, the foregoing implies a specific case where

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