ABCD is a parallelogram in which P and Q are midpoints of opposite sides AB and CD (see the given figure). If AQ intersects DP at S and BQ intersects CP at R. Show that:

APCQ is a parallelogram.

ABCD is a parallelogram in which P and Q are midpoints of opposite sides AB and CD (see the given figure). If AQ intersects DP at S and BQ intersects CP at R. Show that: DPBQ is a parallelogram.

In a parallelogram ABCD, E and F are the midpoints of the sides AB and DC respectively. Show that the line segments AF and EC trisect the diagonal BD.

If P and Q are the middle points of the sides BC and CD of the parallelogram ABCD, then AP+AQ is equal to

E and F are respec­tively the midpoints of equal sides AB and AC of triangle ABC (see the given figure). Show that BF = CE.

In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP= BQ (see the given figure) Show that: AP= CQ

triangle ABC and triangle DBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC (see the given figure). If AD is extended to intersect BC at P, show that, ( i ) triangle ABD = triangle ACD (ii) triangle ABP = triangle ACP (iii) AP bisects angle A as well as angle D. (iv) AP is the perpendicular bisector of BC

In the adjacent figure ABCD is a parallelogram P and Q are the midpoints of sides AB and DC respectively. Show that PBCQ is also a parallelogram.

AB is a line segment. P and Q are points on opposite sides of AB such that each of them is equidistant from the points A and B (see the given figure). Show that the line PQ is the perpendicular bisector of AB.

In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP= BQ (see the given figure) Show that: DeltaAQB ~= DeltaCPD

In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP= BQ (see the given figure) Show that: DeltaAPD ~= DeltaCQB