ABC is an isoscles triangle and the straight line XY is parallel to BC intersects the other two sides at the points X and Y. Prove the points B,C,X,Y are cyclic.

Two circles intersect each other at the points G and H . A straight line is drawn through the point G which intersect two circles at the points P and Q and the straight line through the point H parallel to PQ intersects the two circles at the points R and S . Prove that PQ=RS .

A straight line intersects one of the two concentric circles at the points A and B and other at the points C and D. Prove that AC = BD.

The centre of two circle are P and Q they intersect at the points A and B. The straight line parallel to the line segment PQ through the point A intersects the two circles at the point C and D Prove that CD = 2PQ

In an isosceles right-angled triangle ABC, /_B=90^(@) . The bisector of /_BAC intersects the side BC at the point D. Prove that CD^(2) =2BD^(2)

Two circle intersect each other at A and B and a straight line parallel to AB intersects the circles at C,D,E,F. Prove that CD = EF.

A straight line parallel to any side of any triangle divides other two sides (or the extended two sides)__________ .

The straight line parallel to the side BC of DeltaABC intersects the sides AB and AC at the points D and E respectively. Then (AB)/(BD)=(AC)/(CE) .

In DeltaABC , a straight line parallel to BC intersects the sides AB and AC at the points L and M . If the length of AM be thrice of the length of AL , then BL:CM=

ABC is a triangle and through A, B, C lines are drawn parallel to BC, CA and AB respectively intersecting at P, Q and R. Prove that the perimeter of DeltaPQR is double the perimeter of DeltaABC.

Two parallel straight lines intersect three concurrent straight lines at the points A , B , C and X,Y,Z respectively. Prove that AB : BC=XY : YZ .