In Fig. 14.100, ABCD and PQRC are rectangles and Q is the mid-point of AC. Prove that

(i) DP = PC (ii) PR = AC.

ABC is a triangle. D is a point on AB such that AD=AB and E is a point on AC such that AE= AC. Prove that DE=BC.

In Fig. 14.99, ABCD is a parallelogram in which P is the mid-point of DC and Q is a point on AC such that CQ= AC. If PQ produced meets BC at R, Prove that R is a mid-point of BC.

ABCD is a parallelogram, E and F are the mid-points of AB and CD respectively. GH is any line intersecting AD,EF and BC at G, P and H respectively. Prove that GP=PH.

BM and CN are perpendiculars to a line passing through the vertex A of a triangle ABC. If L is the mid-point of BC, prove that LM=LN.