If in a triangle ABC, medians from A and B are perpendicular to each other, then show that a^{2}+b^{2}=5c^{2}
The intersection of the medians divides each median in a 2:1 ratio, with the longer portion toward the appropriate triangle vertex and the shorter portion toward the side being bisected. Consider the distance from A to the median intersection 2x, and the remainder of that median is then x. The distance from B to the median intersection is 2y and the remainder of that median is then y.
By the right triangles, using the Pythagorean Theorem, (a/2)^2 = 4*y^2 + x^2, or a^2 = 16*y^2 + 4*x^2. Similarly, b^2 = 16*x^2 + 4*y^2.
Also, by its right triangle, c^2 = 4*y^2 + 4*x^2.
Adding, a^2 + b^2 = 20*x^2 + 20*y^2. This is 5*c^2.

Posted by Charlie
on 20080614 12:04:37 