Problem:
The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths 23,5, and 37, as shown, is nmp, where m, n, and p are positive integers, m and n are relatively prime, and p is not divisible by the square of any prime. Find m+n+p.
Solution:
Let the given triangle have vertices (0,0),(5,0), and (0,23) in the coordinate plane. Then the hypotenuse of the triangle lies on the line 23x+5y=103. Suppose the equilateral triangle has side length s with a side that has endpoints (scosθ,0) and (0,ssinθ) for some θ between 0 and 2π. The midpoint of that side is 2s(cosθ,sinθ). The altitude of the equilateral triangle to this midpoint must have slope sinθcosθ and have length 23s, so there is a vertex of the equilateral triangle at 2s(cosθ+3sinθ,sinθ+3cosθ ). Because this vertex lies on the line 23x+5y=103, it follows that 2s⋅23(cosθ+3sinθ)+2s⋅5(sinθ+3cosθ)=103, from which
s=73cosθ+11sinθ203.
Let A=(73)2+112=267. Then there is an angle α such that cosα=A73 and
s=cosαcosθ+sinαsinθA203=cos(α−θ)A203
Thus the minimum possible side length occurs when θ=α and s=A203=67103. The minimum area for the equilateral triangle is 43s2=67753. The requested sum is 75+67+3=145.
