Two Homotheties in a Parallelogram
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A Mathematical Droodle

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Copyright © 1996-2017 Alexander Bogomolny

In parallelogram ABCD, M lies on the diagonal BD, P on CD, and N on AD so that MP||AD and MN||AB. Parallelogram NMPD slides to a position ATUS. Then point U lies on the diagonal AC.

Indeed, by construction, parallelograms ABCD and NMPD are homothetic at D and, hence, similar. It follows that parallelograms ATUS and ABCD are also similar and, therefore homothetic, now at A. Since, C is the image of U under the latter homothety, points A, U, and C are collinear.

This is the content of Euclid VI.24 and VI.26:

Proposition VI.24

In any parallelogram the parallelograms about the diameter are similar both to the whole and to one another.

Proposition VI.26

If from a parallelogram there be taken away a parallelogram similar and similarly situated to the whole and having a common angle with it, it is about the same diameter with the whole.

Simple but strangely useful fact!

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