Two Homotheties in a Parallelogram
What is this about?
A Mathematical Droodle

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

What if applet does not run?


|Activities| |Contact| |Front page| |Contents| |Geometry|

Copyright © 1996-2018 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!

|Activities| |Contact| |Front page| |Contents| |Geometry|

Copyright © 1996-2018 Alexander Bogomolny