In an isosceles ` A B C` , the base `A B` is produced both the ways to `P` and `Q` such that `A PxxB Q=A C^2dot` Prove that ` A P C B C Q` .

In an isosceles triangle A B C , where CA = CB the base A B is produced both the ways to P and Q respectively, such that A PxxB Q=A C^2 . Prove that triangle APC is similar to triangle BCQ.

In Figure, P , is any point on the chord B C of a circle such that A B=A P . Prove that C P=C Q .

A B C is a triangle in which A B=A C\ a n d\ P is a point on A C . Through C a line is drawn to intersect B P produced at Q such that /_A B Q=\ /_A C Q . Prove that: /_A Q C=90^@+1/2/_B A 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 C D is a parallelogram. P is a point on A D such that A P=1/3\ A D\ a n d\ Q is a point on B C such that: C Q=1/3B Cdot Prove that A Q C P is a parallelogram.

In A B C and P Q R Figure, A B=P Q ,B C=Q R and C B and R Q are extended to X and Y respectively and /_A B X =/_P Q Ydot Prove that A B C~=P Q Rdot Figure

In A B C and P Q R Figure, A B=P Q ,B C=Q R and C B and R Q are extended to X and Y respectively and /_A B X = /_P Q Ydot Prove that A B C~=P Q Rdot Figure

In Figure, A P || B Q\ || C R . Prove that a r\ ( A Q C)=\ a r\ (P B R)

In figure, P is the mid-point of B C , ,Q is the mid-point of A P , such that B Q produced meets A C at Rdot Prove that 3RA= CA

In A B C , P divides the side A B such that A P : P B=1:2 . Q is a point in A C such that P Q B C . Find the ratio of the areas of A P Q and trapezium B P Q C .