Question

Let ABCD be a square. Locate points P, Q, R, S on the sides AB, BC, CD, DA respectively such that AP = BQ = CR = DS. Prove that PQRS is a square.

Answer 1

Given: Square ABCD with points P, Q, R and S on sides AB, BC, CD and AD respectively such that AP = BQ = CR = DS

To Prove: PQRS is a square

Proof:

Let the side of the square ABCD be a and let AP = BQ = CR = DS = x.

In triangle PBQ, we have:

In triangle QRC, we have:

In triangle RDS, we have:

In triangle ASP, we have:

From (1), (2), (3) and (4), we get:

Thus, PQRS is a square.