In a ` A B C ,\ E\ a n d\ F` are the mid-points of `A C\ a n d\ A B` respectively. The altitude `A P` to `B C` intersects `F E\ ` at `Q`. Prove that `A Q=Q P`.

In square A B C D ,\ P\ a n d\ Q are mid-point of A B\ a n d\ C D respectively. If A B=8c m\ a n d\ P Q\ a n d\ B D intersect at O , then find area of O P B

A B C is a triangle in which D is the mid-point of B C ,\ E\ a n d\ F are mid-points of D C\ a n d\ A E respectively. If area of A B C is 16\ c m^2, find the area of D E F

A B C D is a parallelogram, E\ a n d\ F are the mid-points of A B\ a n d\ C D respectively. G H is any line intersecting A D ,\ E F\ a n d\ B C at G ,\ P\ a n d\ H respectively. Prove that G P=P H

In Figure, X\ a n d\ Y are the mid-points of A C\ a n d\ A B respectively, Q P || B C\ a n d\ C Y Q\ a n d\ B X P are straight lines. Prove that a r\ (\ A B P)=\ a r\ (\ A C Q) .

A B C is an isosceles triangle in which A B=A C . If D\ a n d\ E are the mid-points of A B\ a n d\ A C respectively, Prove that the points B ,\ C ,\ D\ a n d\ E are concyclic.

A D\ a n d\ B E are respectively altitudes of A B C such that A E=B D . Prove that A D=B E .

In Figure, A B C D is a parallelogram. E\ a n d\ F are the mid-points of the sides A B\ a n d\ C D respectively. Prove that the line segments A F\ a n d\ C E trisect (divide into three equal parts) the diagonal B D .

ABCD is a rhombus and P ,Q ,R ,S are the mid-points of A B ,B C ,C D ,D A respectively. Prove that P Q R S is a rectangle.

A B C D is a trapezium with parallel sides A B=a\ a n d\ D C=bdot If E\ a n d\ F are mid-points of non-parallel sides A D\ a n d\ B C respectively, then the ratio of areas of quadrilaterals A B F E\ a n d\ E F C D is: a : b (b) (a+3b):(3a+b) (3a+b):(a+3b) (d) (3a+b):(3a+b)

A B C D is a rhombus and P ,\ Q ,\ R ,\ S are the mid-points of A B ,\ B C ,\ C D ,\ D A respectively. Prove that P Q R S is a rectangle.