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 the given triangle P, Q and R are the mid points of sides AB, BC and AC respectively. Prove that triangle PQR is similar to triangle ABC.

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 .

In a triangle \ A B C , If L\ a n d\ M are points on A B\ a n d\ A C respectively such that L M || B C . Prove that: a r\ (triangle \ L O B)=a r\ (triangle \ M O C)

ABC is an isosceles triangle with A B =\ A C . Draw A P_|_B C to show that /_B =/_C.

A B C is an isosceles triangle with A C=B C . If A B^2=2\ A C^2 , prove that A B C is right triangle.

The line x+y=p meets the x- and y-axes at Aa n dB , respectively. A triangle A P Q is inscribed in triangle O A B ,O being the origin, with right angle at QdotP and Q lie, respectively, on O Ba n dA B . If the area of triangle A P Q is 3/8t h of the are of triangle O A B , the (A Q)/(B Q) is equal to (a) 2 (b) 2/3 (c) 1/3 (d) 3

ABC is an isosceles triangle with AC = BC. If A B^2=2A C^2 , prove that ABC is a right triangle.

A plane meets the coordinate axes at A, B and C respectively such that the centroid of triangle ABC is (p, q, r). The equation of the plane is

If a cosA=b cos B , then prove that either the triangle is isosceles or right triangle.

In triangle ABC, the sides AB and AC are equal. If P is a point on AB and Q is a point on AC such that AP = AQ, prove that : (i) triangle APC and AQB are congruent. (ii) triangle BPC and CQB are congruent.