How to prove the similarity of two rectangles?

Question

The task is this: On the ABCD rectangles AB,BC,CD,DA sides we took up P,Q,R,S points, so PR and QS are perpendicular to each other. Let's prove that the middle points of SP,PQ,QR,RS segments form a rectangle that is similar to the ABCD rectangle.

I know how to prove it when the P,Q,R,S points are the middle points of the ABCD rectangle, but I do not know how to prove it generally.

Could you please show me how to prove it?

Answer 1

Hint:

• Let $KLMN$ be a rectangle formed by the middle points of $SP$, $PQ$, $QR$, and $RS$.
• First, prove that ratio of lengths of sides $KL : LM$ is the same as ratio of $PR : QS$.
• Draw two segments parallel to $PR$ and $QS$ respectively, that go through some corner of the original rectangle.
• Observe that by similarity of right triangles, the ratio of these two segments (and hence the ratio of $PR : QS$) is the same as $AB : BC$.

                                                  

I hope this helps $\ddot\smile$