ABCD is a parallelogram in which P is the midpoint of DC and Q is a point on AC such that CQ = 1/4 AC. If PQ produced meets BC at R, prove that R is the midpoint of BC.

Question

Philoid · Accepted Answer

We have the diagram as

Given: DP = PC &

CQ = (1/4)AC …(i)

To Prove: CR = RB

Proof: Join B to D

As diagonals of a parallelogram bisect each other at S.

…(ii)

Dividing equation (i) by (ii), we get

⇒

⇒ CQ = CS/2

⇒ Q is the midpoint of CS.

According to midpoint theorem in ∆CSD, we have

PQ ∥ DS

Similarly, in ∆CSB, we have

QR ∥ SB

Also, given that CQ = QS

We can conclude that, by the converse of midpoint theorem, CR = RB.

That is, R is the midpoint of CB.

Hence, proved.