In a triangle A B C, B A C=90^ . Let D be the point on B C such that A B+B D=A C+C D . Suppose B D: D C=2: 1 . If A C A B = m+ p n , where m, n are relatively prime positive integers and p is a prime number, determine the value of m+n+p .

Step-by-step solution

A B+B D=A C+C D aligned & y+2 x=A C+x & A C=x+y & (x+y)^2+y^2=(3 x)^2 & x^2+2 y^2+2 x y=9 x^2 & 8 x^2-2 y^2-2 x y=0 & 4 a^2-a-1=0 & [ array ll & a= x y array ] & a= 1 1+16 8 & a= 1 17 8 & A C A B = x y +1=a+1 & A C A B = 1+ 17 8 +1 & A C A B = 9+ 17 8 & m+n+p=34 aligned