ABCD is a parallelogram in which P is the midpoint of DC and Q is a point on AC such that CQ = AC. If PQ produced meets BC at R, prove that R is the midpoint of BC.

Answer:

 

 

 

We know that the diagonals of a parallelogram bisect each other.

Therefore,

CS = 

AC …(i)

Also, it is given that CQ =

AC …(ii)

Dividing equation (ii) by (i),

Q is the midpoint of CS

According to midpoint theorem in ∆CSD

PQ || DS

If PQ || DS,

QR || SB

In ∆ CSB, Q is midpoint of CS and QR ‖ SB.

Applying converse of midpoint theorem,

Conclusion – R is the midpoint of CB.

Hence, proved.