`/_\PBC` and `/_\ QBC` are two isosceles triangles on the same side of the some base. Show that the PQ is the right bisector of BC.

/_PBC , /_QBC , /_RBC are three isosceles triangles on the same base BC. Show that P, Q, R are collinear.

triangleBAC and triangleBDC are two right-triangle on the same side of the base BC.If AC ans DB intersect each other at a point P,show that APxxPC=DPxxPB .

If one angle of a triangle is equal to one angle of another triangle and the bisectors of these equal angles divide the opposite side in the same ratio then show that the triangle are similar.

In an isosceles triangle ABC if AC=BC and AB^2 = 2AC^2 , prove that /_C is a right angle.

prove that two isosceles triangles are similar if their vertical angles are equal. (it is supposed that the angle on the base of a triangle are equal).

If cosines of two angles of a triangle are proportional to the opposite sides, show that the triangle is isosceles

ABC is an isosceles triangle with AC=BC.If AB^2=2AC^2 .prove that ABC is a right triangle.

Two triangles ABC and DBC are in the same side of the common base BC. Lines drawn parallel to BA and BD from any point E on BC intersect AC and DC at the points P and Q respectively. Show that PQ is parallel to AD.

If cosines of two angles of a triangle are inversely proportional to the opposite sides, show that the triangle is either isosceles or right-angled.

A and D are two points on the perpendicular bisector of BC lying on the same side of BC. Prove that /_ABD = /_ACD .