Distances to the sides of an equilateral triangle

Question

Take an equilateral triangle and any point P inside it. Drop perpendiculars from P to the three sides and measure their lengths.

Prove that the sum of the three distances does not depend on where P is, and say what it equals. Then explain what happens for a triangle that is not equilateral.

Accepted Answer

The sum of the three distances equals the height of the triangle, wherever P is. Areas. Let the side length be s and the three distances be a, b and c. Joining P to the three corners cuts the big triangle into three triangles with bases s, s, s and heights a, b, c. Their areas are (1/2) s a, (1/2) s b and (1/2) s c, and they add up to the area of the whole triangle, which is (1/2) s h where h is the height. So (1/2) s (a + b + c) = (1/2) s h, giving a + b + c = h. Check with two positions. At the centre, the three distances are each h/3, summing to h. At a corner, two distances are 0 and the …

Distances to the sides of an equilateral triangle

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