Let A B C be a triangle let D be a point on the segment B C such that A D=B C . Suppose C A D=x^ , A B C=y^ and A C B=z^ and x, y, z are in an arithmetic progression in that order where the first term and the common difference are positive integers. Find the largest possible value of A B C in degrees.

Step-by-step solution

A D y = B D (x+y+z) A D z = C D x aligned & (x+y+z) y + x z =1 & (3 y) y + (y-d) (y+d) =1 & (y-d) (y+d) =-2 2 y & (3 y+d)=0 3 y+d=180^ , 360^ & y & d are integers d is multiple of 3 & If 3 y+d=180^ y_ =59^ & 3 y+d=360^ y_ =119^ (if y is obtuse, then z must be acute) & Only 1^ st case is possible y_ =59^ aligned