In the following figure, OP, OQ and OR are drawn perpendiculars to the sides BC, CA and AB repectively of triangle ABC. Prove that :
`AR^(2)+BP^(2)+CQ^(2)=AQ^(2)+CP^(2)+BR^(2)`

D, E and F are the mid-points of the sides BC, CA and AB respectively of triangle ABC. Prove that: BDEF is a parallelogram.

D, E and F are the mid-points of the sides BC, CA and AB respectively of triangle ABC. Prove that: area of BDEF is half the area of Delta ABC.

In the adjoining figure, DeltaPQR is formed by the sides PQ, QR and RP which are drawn parallel to sides AB,BC and CA respectively of DeltaABC. Prove that " "PQ+QR+RP=2(AB+BC+CA).

If G be the centroid of a triangle ABC, prove that, AB^2 + BC^2 + CA^2 = 3(GA^2 + GB^2 + GC^2)

In a triangle ABC, AB = BC, AD is perpendicular to side BC and CE is perpendicular to side AB. Prove that : AD = CE.

In an equilateral triangle ABC, BE is perpendicular to side CA. Prove that: AB^(2) + BC^(2) +CA^(2) = 4BE^(2)

In the following figure, AD is perpendicular to BC and D divides BC in the ration 1 : 3. Prove that : 2AC^(2)=2AB^(2)+BC^(2)

In triangle ABC, AB=AC and BD is perpendicular to AC. Prove that : BD^(2)-CD^(2)=2CDxxAD

In the figure AB = BC and AD is perpendicular to CD. Prove that : " "AC^(2)=2.BC.DC

In the adjoining figure D, E and F are the mid-points of the sides BC, CA and AB respectively of Delta ABC . Prove that: (i) square BDEF is a parallelogram (ii) area of Delta DEF = (1)/(4) xx " area of " Delta ABC (iii) square BDEF = (1)/(2) xx " area of " Delta ABC