MEDIUM
Olympiad
IMPORTANT
Earn 100

The diagonals AC and BD of a cyclic quadrilateral ABCD meet at right angles in E. Prove that EA2+EB2+EC2+ED2=4R2, where R is the radius of the circumscribing circle.

Important Questions on Geometry

HARD
Olympiad
IMPORTANT
Suppose ABCD is a rectangle and P,Q,R,S are points on the sides AB,BC,CD,DA respectively. Show that PQ+QR+RS+SP>2AC.
MEDIUM
Olympiad
IMPORTANT
Let P be an interior point of an equilateral triangle ABC such that AP2=BP2+CP2. Prove that BPC=150°.
EASY
Olympiad
IMPORTANT

Let ABC be a triangle and ha be the altitude through A. Prove that b+c2a2+4ha2 (As usual a,b,c denote the sides BC,CA,AB respectively)

HARD
Olympiad
IMPORTANT
Let P be an interior point of a triangle ABC and let BP and CP meet AC and AB in E and F respectively. If BPF=4,BPC=8 and CPE=13, find AFPE. (Here, denotes the area of a triangle or a quadrilateral as the case may be)
MEDIUM
Olympiad
IMPORTANT

Suppose ABCD is a cyclic quadrilateral inscribed in a circle of radius one unit. If AB·BC·CD·DA4, then prove that ABCD is a square.