∆ABC and ∆DBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC (see figure). If AD is extended to intersect BC at P, show that:

(i)    ∆ABD ≅ ∆ACD
(ii)    ∆ABP ≅ ∆ACP
(iii)    AP bisects ∠A as well as ∠D
(iv)    AP is the perpendicular bisector of BC.

AD is an altitude of an isosceles triangle ABC in which AB = AC. Show that

(i)    AD bisects BC
(ii)    AD bisects ∠A.

Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of triangle PQR (see figure). Show that:

(i) ∆ABM ≅ ∆PQN
(ii) ∆ABC ≅ ∆PQR.

66. BE and CF are two equal altitudes of a triangle ABC. Using RHS congruence rule, prove that the triangle ABC is isosceles.

The diagonals PR and QS of a quadrilateral PQRS intersect each other at O. Prove that

(i) PQ + QR + RS + SP >  PR + QS
(ii) PQ + QR + RS + SP <  2 (PR + QS)