Carnival triangles

Here's a cute little geometric factoid that has something to do with one of the posts over at the 33rd Carnival of Mathematics. I'll leave it as a puzzle which post it belongs to...

Let \( ABC \) be any triangle in the Euclidean plane, and \( AD \) be any line. Form points \( A' \), \( B' \), and \( C' \) as the perpendicular projections of \( A \) onto \( BC \), \( B \) onto \( AD \), and \( C \) onto \( AD \) respectively. Then triangles \( ABC \) and \( A'B'C' \) are similar.

(Hint: in the post I have in mind, \( ABC \) is isosceles and \( A' \) is the midpoint of \( BC \).)