# Properties of Flank Triangles - Proof with an Asymmetric Rotation

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.

\(EG = 2\cdot AM\).

The applet below illustrates one of the solutions. Use the slider at the top of the applet to rotate \(\triangle AEG\) \(90^{\circ}\) clockwise around \(O\) - the center of square \(ABDE\).

### Proof

The rotation around \(O\) maps \(E\) to \(A\), \(A\) to \(B\), and \(G\) to \(G'\) such that the quadrilateral \(ABG'C\) is a parallelogram. (This is because \(BG'\parallel AC\) and \(BG' = AG = AC\).) So naturally, \(AM\) is half the diagonal \(AG' = EG\).

Also, by the construction, \(AG'\perp EG\) and the same holds for \(AM\): \(AM\perp EG\).

### References

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

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