Reach for the Summit - M-S5-A1

Geometry Level 3

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

If the circumcenters of AMN\triangle AMN and ABC\triangle ABC are D(x1,y1)D(x_1,y_1) and E(x2,y2)E(x_2,y_2) respectively, find 1000(x1+2y1+3x2+4y2)\lfloor 1000(x_1+2y_1+3x_2+4y_2)\rfloor.

Hint: Find the relationship of ADAD and AEAE.

Reach for the Summit problem set - Mathematics


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