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Mutually Perpendicular Medians (Posted on 2008-06-14) Difficulty: 3 of 5
If in a triangle ABC, medians from A and B are perpendicular to each other, then show that a2+b2=5c2

See The Solution Submitted by Praneeth    
Rating: 2.0000 (3 votes)

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Solution solution | Comment 1 of 4

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 2008-06-14 12:04:37
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