SOLUTION: Point P is inside equilateral triangle ABC such that the altitudes from P to AB, BC, and CA have lengths 5, 6, and 7 respectively. What is the area of triangle ABC?

Question 1087861: Point P is inside equilateral triangle ABC such that the altitudes from P to AB, BC, and CA have lengths 5, 6, and 7 respectively. What is the area of triangle ABC?

Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
PA, PB, PC splits triangle ABC into three triangles
:
let s be the length of a side of equilateral triangle ABC
:
5s/2 + 6s/2 + 7s/2 = s^2 * square root(3) / 4 (area of equilateral triangle)
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divide both sides of = by s
:
5/2 + 6/2 +7/2 = (s * square root(3)) / 4
:
multiply both sides of = by 4
:
10 + 12 + 14 = s * square root(3)
:
s = 36 / square root(3)
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multiply the numerator and denominator by square root(3)
:
s = 36 * square root(3) / 3 = 12 * square root(3)
:
s^2 = (12 * square root(3))^2 = 144 * 3 = 432
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area of ABC = (432 * square root(3) / 4)
:
area of ABC = 108 * square root(3)
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