Viviani's Theorem proves this for isosceles triangles. If you take a look at
http://www.cut-the-knot.org/triangle/VivianiByVectors.shtml
where it is proved for equilateral triangles, you should be able to see that it applies equally well to arbitrary triangles.
If not, you may be able to find a suitable extension to Viviani's Theorem.
in an equilateral triangle ABC,O is any point in the interior of the triangle.From
O perpendiculars are drawn to the sides.prove that sum of these perpendiculars is constant for any triangle
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