Set up the given triangle on x-y coordinates with right angle at (0,0). So the two vertices are at (5,0) and (0,2[tex]sqrt{x} n]{3} [/tex])
let (a,0) and (0,b) be two vertices of the equilateral triangle. So the third vertex must be at [tex](\frac{a+\sqrt{3}b}{2} , \frac{b+\sqrt{3}a}{2} )[/tex]
for a pt (x,y) on line sx+ty=1, the minimum of [tex] \sqrt{x^{2} + {y^{2} } [/tex]
equals to [tex] \frac{1}{ \sqrt{ s^{2} + t^{2} } } [/tex]
smallest value happens at [tex] \frac{10 \sqrt{3} }{67} [/tex]
so area is [tex] \frac{75\sqrt{3}}{67} [/tex]
hence m=75, n=67, p=3
m+n+p = 75+67+3 = 145
