ABC is and isosceles triangle whose ∠C is right angle. If D is any point on AB, then let us prove that, AD² + DB² = 2CD²

I have drawn a triangle ABC whose height is AD. If AB>AC, let us prove that, AB²-AC = BD²-CD²

In ΔABC I have drawn two perpendiculars from two vertices B and C on AC and AB (AC>AB) which are intersected each other at the point P. Let us prove that, AC² + BP² = AB² + CP²

In the triangle ABC, A is right angle, if CD is a median, let us prove that, BC² = CD² + 3AD²

From a point O within a tringle ABC, I have drawn the perpendiculars OX, OY and OZ on BC, CA and AB respectively. Let us prove that, AZ² + BX² + CY² = AY² + CX² + BZ²