Geometry : 106

117: Random point in an equilateral triangle

A point P is chosen at random inside an equilateral triangle of side length 1.  Find the expected value of the sum of the (perpendicular) distances from P to the three sides of the triangle.

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We will show that the sum of the perpendicular distances from an arbitrary point P inside the equilateral triangle to the three sides of the triangle is a constant.

Draw lines from P to each of the vertices of the triangle.  These lines divide the equilateral triangle into three triangles.  Each of these triangles has as its base one side of the equilateral triangle and as its height the perpendicular distance from P to that side.  Let those perpendicular distances be h ₁, h₂, and h₃.

The area of a triangle is equal to ½ × base × perpendicular height.
Hence the area of each internal triangle is, respectively, ½h₁, ½h₂, and ½h₃.

The height of the equilateral triangle is readily found by dropping a perpendicular from a vertex to the opposite side, and applying Pythagoras' Theorem .
We find that the height is /2, and so the area of the equilateral triangle is  /4.

The area of the equilateral triangle is equal to the sum of the areas of the three internal triangles.
Hence /4 = ½h₁ + ½h₂ + ½h₃, and so h₁ + h₂ + h₃ = /2.

Since the sum of the perpendicular distances is a constant, the expected value of the sum of the perpendicular distances from P to the three sides of an equilateral triangle of side length 1 is  /2.