The right bisector of AB and AC of the triangle ABC intersect at 0. Prove that `OA = OB = OC`.

the altitude AM and BN of a triangle ABC intersects at P. prove that triangleAPN and triangleBPM are similar.

If PB and AQ are the medians of a ∆ABC right angled at C. Prove that 4BP^2 = 4 BC^2 + AC^2

If PB and AQ are the medians of a ∆ABC right angled at C. Prove that 4AQ^2 = 4 AC^2 + BC^2

The perpendicular from C on the side AB of a triangle ABC intersect AB at D and BD=mAD . Prove that (m+1)BC^2=(m+1)AC^2+(m-1)AB^2 .

If PB and AQ are the medians of a ∆ABC right angled at C. Prove that 4 (AQ^2 + BP^2) = 5 AB^2

ABC is an isosceles triangle right angled at C.Prove that AB^2=2AC^2 .

E is the middle point of the median AD of /_\ABC . BE intersects AC at F. Prove that AF=1/3AC .

If angleC of the isosceles triangle ABC is measure 90^@ , then prove that AB^2=2AC^2 .

If D is the middle point of the side BC of the triangle ABC, prove that AB^2 + AC^2 = 2(AD^2 + DC^2) .

D and E are points on the sides CA and CB respectively of a triangle ABC right angled at C.Prove that, AE^2+BD^2=AB^2+DE^2 .