In Figure, `A B C` is an isosceles triangle in which `A B=A CdotC P A B` and `A P` is the bisector of exterior `/_C A D` of ` A B C` . Prove that `/_P A C=/_B C A` and (ii) `A B C P` is a parallelogram.

In figure A B=A C\ a n d\ C P||B A and A P is the bisector of exterior /_C A D of A B C . Prove that /_P A C=/_B C A (ii) A B C P is a parallelogram

ABC is an isosceles triangle in which A B" "=""A C . AD bisects exterior angle PAC and C D" "||""A B . Show that (i) /_D A C" "=/_B C A and (ii) ABCD is a parallelogram.

In Figure, A N and C P are perpendicular to the diagonal B D of a parallelogram A B C Ddot Prove that : A D N~= C B P (ii) A N=C P

A B C\ a n d\ D B C are two isosceles triangles on the same bas B C and vertices A\ a n d\ D are on the same side of B C . If A D is extended to intersect B C at P , show that A B D\ ~= A C D (ii) A B P\ ~= A C P

A B C D is a cyclic quadrilateral whose diagonals A C\ a n d\ B D intersect at Pdot If A B=D C , prove that: P A B\ ~= P D C P A=P D\ a n d\ P C=P B (iii) A D B C

A B C and D B C are two isosceles triangles on the same base B C and vertices A and D are on the same side of B D . If A D is extended to intersect B C at P , show that A B D~=A C D (ii) A B P~=A C P A P bisects /_A as well as /_D A P is perpendicular bisector of B Cdot

A B C D is a cyclic quadrilateral whose diagonals A Ca n dB D intersect at P . If A B=D C , Prove that : P A B~= P D C P A=P Da n dP C=P B A D B C

If P is any point in the interior of a parallelogram A B C D , then prove that area of the triangle A P B is less than half the area of parallelogram.

In A B C , /_A is obtuse, P B_|_A C and Q C_|_A B . Prove that: A BxxA Q=A CxxA P (ii) B C^2=(A CxxC P+A BxxB Q)

In Figure, A B C D is a parallelogram in which P is the mid-point of D C\ a n d\ Q is a point on A C such that C Q=1/4A Cdot If P Q produced meets B C\ a t\ R , prove that R is a mid-point of B C