Here we will prove that if each diagonal of a quadrilateral divides it in two triangles of equal area then prove that the quadrilateral is a parallelogram.
Given: PQRS is a quadrilateral whose diagonals PR and QS cut at O such that
ar(∆PQR) = ar(∆PSR), and
ar(∆PQS = ar(∆QRS).
To prove: PQRS is a parallelogram.
1. ar(PQR) = ½ ar(quadrilateral PQRS).
2. ar(PSQ) = ½ ar(quadrilateral PQRS).
3. ar(PQR) = ar(PSQ)
4. SR ∥ PQ.
5. QR ∥ PS.
6. PQRS is a parallelogram. (Proved)
1. Given, ar(∆PQR) = ar(∆PSR).
2. Given, ar(∆PSQ) = ar(∆QRS).
3. From statements 1 and 2.
4. ∆PQR and ∆PSQ are of equl area on the same base PQ.
5. Similarly with PS as base.
6. From statements 4 and 5.