Question

In ΔABC, the medians BE and CD are produced to the points P and Q respectively such that BE = EP and CD = DQ. Prove that: A is the mid-point of PQ.

Answer 1

In ΔBDC and ΔADQ,
CD = DQ ....(given)
∠BDC = ∠ADQ ....(vertically opposite angles)
BD = AD ....(D is the mid-point of AB)
∴ ΔBDC ≅ ΔADQ   
⇒ ∠DBC = ∠DAQ  (c.p.c.t)....(i)
And, BC = AQ         (c.p.c.t)....(ii)
Similarly, we can prove ΔCEB ≅ ΔAEP
⇒ ∠ECB = ∠EAP   (c.p.c.t)....(iii)
And, BC = AP         (c.p.c.t)....(iv)

From (ii) and (iv),
AQ = AP
⇒ A is the mid-point of PQ.