Question

In ΔABC, BE and CF are medians. P is a point on BE produced such that BE = EP and Q is a point on CF produced such that CF = FQ. Prove that: A is the mid-point of PQ.

Answer 1

Since BE and CF are medians, F is the mid-point of AB and E is the mid-point of AC. Now, the line joining the mid-point of any two sides is parallel and half of the third side, we have In ΔACQ,

EF || AQ and EF = `(1)/(2)"AQ"` ....(i)

In ΔABP,

EF || AP and EF = `(1)/(2)"AP"` ....(ii)

From (i) and (ii)

`"EF" = (1)/(2)"AQ" and "EF" = (1)/(2)"AP"`

⇒ `(1)/(2)"AQ" = (1)/(2)"AP"`
⇒ AQ = AP
⇒ A is the mid-point of QP.