In any triangle PQR, K and M are the mid-points of PQ and QR respectively. If E be any point on the side PR and KE//QR. Prove that: QK = ME.

Given,

1. PK = KQ
2. QM = MR
3. KE // QR

To Prove: QK = ME

Proof:

1. KE bisects PR       (Line passing through the midpoint of any one side    of a triangle, (PR in this case) and parallel to the third side(KE//QR, in this case), bisects the third side)
2. ME = PQ/2            (Line joining the mid points, M and E(in this case), of any two sides of a triangle, QR and MR(in this case), is half of and parallel to the third side)
3. QK = PQ/2            (K is the mid-point of PQ)
4. QK = ME               (From 2 and 3)