Question

Prove that the quadrilateral that is formed by joining the midpoints of the sides of a square is a square.

(Theorem : The straight line segment through the midpoints of two sides of a triangle is parallel to the third side and equal in length to half of it.)

Answer 1

Imagine the square has side length 2a. The midpoints of the sides are length a from each corner and the hypotenuses of the right-angled triangles at each corner all have the same length a√2. The quadrilateral is therefore a rhombus. The right-angled triangles at the corners are isosceles making the angle between each hypotenuse and each side 45 degrees. The interior angles of the rhombus is 180-45-45=90 degrees, so the rhombus is a square.