Two triangles `B A Ca n dB D C` , right angled at `Aa n dD` respectively, are drawn on the same base `B C` and on the same side of `B C` . If `A C` and `D B` intersect at `P ,` prove that `A P*P C=D P*P Bdot`

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\ 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 P bisects " "/_A as well as /_D and A P is the perpendicular bisector of B C

In a triangle A B C , let P and Q be points on AB and AC respectively such that P Q || B C . Prove that the median A D bisects P Q .

A B is a diameter of a circle C(O ,\ r)dot Chord C D is equal to radius O Cdot If A C\ a n d\ B D when produced intersect at P , prove that /_A P B is constant.

P and Q are points on the sides C A and C B respectively of A B C , right-angled at C . Prove that A Q^2+B P^2=A B^2+P Q^2 .

P is a point on the side B C of a triangle A B C such that A B=A P . Through A\ a n d\ C ,\ lines are drawn parallel to B C\ a n d\ P A , respectively, so as to intersect at D as shown in Figure. Show that A B C D is a cyclic quadrilateral.

If A B C and A M P are two right triangles, right angled at B and M respectively such that /_M A P=/_B A C . Prove that (C A)/(P A)=(B C)/(M P)

In triangle A B C ,\ P\ a n d\ Q are respectively the mid-points of A B\ a n d\ B C and R is the mid-point of A P . Prove that: a r\ ( triangle P B Q)=\ a r\ (triangle \ A R C)

In figure ABC and AMP are two right triangles, right angles at B and M respectively. Prove that(i) DeltaA B C~ DeltaA M P (ii) (C A)/(P A)=(B C)/(M P)

In fig.22, A B C is right angled at Adot\ Q\ \ a n d\ R are points on line B C and P is a point such that Q P||A C and R P||A B . Find /_Pdot