`overset(harr)("AB ") and overset(harr)("DC ")` are two parallel lines and a transversal l, intersects `overset(harr)("AB ")` at P and `overset(harr)("DC ")` at R. Prove that the bisectors of the interior angles form a rectangle.

If a transversal intersects two lines such that the bisectors of a pair of corresponding angles are parallel, then prove that the two lines are parallel.

AD is a median of triangleABC . A straight line parallel to BC intersect AB and AC at P and Q respectively. Prove that AD bisects PQ.

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.

In the parallelogram ABCD , AB = 2AD . Prove that the bisectors of the angles angleBAD and angleABC meet at the mid-point of DC at right -angles.

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.

l, m and n are three parallel lines intersected by the transversals p and q at A, B, C and D,E, F such that they make equal intercepts AB and BC on the transversal p. Show that the intercepts DE and EF on q are also equal.

In trapezium ABCD , AB and DC are parallel. A straight line parallel to AB intersects AD and BC at the points E and F respectively. Prove that DE:EA=CF:FB .

ABCD is a cyclic quadrilaterla. Extended AB and DC intersect at P. Prove that PA.PB = PC.PD.

A straight lines L through the origin meets the lines x+y=1 and x+y=3 at P and Q respectively. Through P and Q two straight lines L_1 and L_2 are drawn, parallel to 2x-y=5 and 3x+y=5 respectively. Line L_1 and L_2 intersect at R. Show that the locus of R as L varies is a straight line.

The coordinates of a point referred to rectangular axis overset(harr)(OX)andoverset(harr)(OY) are (4,-3) , state which of the following are the coordinates of the same point when the axis are shifted to the point (-2,5) without rotation ?