geometry: prove sum of perpendicular distances...

[missa0312]

1. Suppose that Triangle ABC is an equilateral triangle and that P is a point in the interior of this triangle. Prove that the sum of the perpendicular distances from P to each of the sides of the triangle is equal to the height oh the triangle.

 

[Denis]

1. Suppose that Triangle ABC is an equilateral triangle and that P is a point in the interior of this triangle. Prove that the sum of the perpendicular distances from P to each of the sides of the triangle is equal to the height oh the triangle.

Keeping it short; I'll assume you'll be able to follow this:

k = sides of triangle ABC, h = height, A = area; then:

h = k sqrt(3) / 2 ; 2h = k sqrt(3) [1]
A = k^2 sqrt(3) / 4 [2]

d,e,f = perpendicular distances; then (using 3 right triangles PAB,PBC,PAC):

A = (dk + ek + fk) / 2 [3]

[2][3]:
k^2 sqrt(3) / 4 = (dk + ek + fk) / 2
k sqrt(3) = 2(d + e + f)
substitute [1]:
2h = 2(d + e + f)
h = d + e + f : height = sum of perpendicular distances

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