What Are Barycentric Coordinates?

(1986 AIME #9) In $\triangle ABC, AB = 425, BC = 450,$ and $AC = 510.$ An interior point $P$ is then drawn, and segments through $P$ are drawn parallel to the sides of the triangle, such that the endpoints of each segment lie on the triangle. If these three segments are of an equal length $d,$ find $d.$