# Two Properties of Flank Triangles - A Proof with Complex Numbers

Bottema's configuration of two squares that share a vertex is naturally embedded into Vecten's configuration of three squares erected on the sides of a triangle. The latter generalizes the first of Euclid's proofs of the Pythagorean theorem, so it rightfully refers to the *Bride's Chair*.

In the discussion of Vecten's configuration, we proved *inter alia* two properties of flank triangles that are a part of Bottema's configuration.

If \(AM\) is a median in \(\triangle ABC\), \(AH\) is an altitude in \(\triangle AEG\), then \(M\), \(A\), \(H\) are collinear.

\(EG = 2\cdot AM\).

### Proof

In geometric problems, synthetic solutions are usually valued more than algebraic or trigonometric. However, complex numbers have a simple geometric meaning. In particular, multiplication by \(i\) is equivalent to a counterclockwise rotation by \(90^{\circ}\) whereas multiplication by \(-i\) is equivalent to a clockwise rotation. Bearing this in mind, the proof appears to truly need no further explanation.

[Honsberger] credits the proof to Boris Pritsker of New York. It appeared in the now defunct journal *Quantum* (May/June 1996, p. 44).

### References

- R. Honsberger,
*Mathematical Diamonds*, MAA, 2003, 63-64

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