Reach for the Summit - M-S5-A1

Geometry Level 3

In $\triangle ABC$ , $A(3,4)$ , $B(-5,0)$ , $M(0,0)$ , $N(2,0)$ , $C$ is a point on the positive $x$ -axis such that $\angle BAM = \angle CAN$ .

If the circumcenters of $\triangle AMN$ and $\triangle ABC$ are $D(x_1,y_1)$ and $E(x_2,y_2)$ respectively, find $\lfloor 1000(x_1+2y_1+3x_2+4y_2)\rfloor$ .

Hint: Find the relationship of $AD$ and $AE$ .

Reach for the Summit problem set - Mathematics