Similar Quadrilaterals: Inspired by a Class Discussion

One day in Geometry II, a topic was over how the lines connecting corresponding vertices of similar triangles were concurrent. The question was raised to us whether this was true of similar quadrilaterals.

Based upon a ruler and pencil sketch, I thought this was not so. I reported my result in class, but it turns out that when this was done with the accuracy of a computer program, this was so. Our instructor and I looked at this after class, and wondered why this was so.

I started out with an irregular quadrilateral, although you can change it to any parallelogram you wish. You can manipulate A, B, C, or D as well as the scaling factor. You will see that the lines connecting corresponding vertices do intersect at one point.

We will discuss why below to mathlet.

Triangles ABD and A'B'D' are similar triangles. That means that the Lines AA', BB', and DD' are all concurrent.

Likewise, Triangles BCD and B'C'D' are similar triangles. That means that the lines BB', CC', DD' are all concurrent.

Both of these similar triangles have Lines BB' and DD' which intersect at a unique point. Lines AA' and CC' intersect at that point as well, so all four such lines must be concurrent.

Darron Steele, Created with GeoGebra