MathDB

12

Part of 1988 AIME Problems

Problems(1)

Cevians in a triangle

Source:

3/19/2006
Let PP be an interior point of triangle ABCABC and extend lines from the vertices through PP to the opposite sides. Let aa, bb, cc, and dd denote the lengths of the segments indicated in the figure. Find the product abcabc if a+b+c=43a + b + c = 43 and d=3d = 3.
[asy] size(200); defaultpen(fontsize(10)); pair A=origin, B=(14,0), C=(9,12), D=midpoint(B--C), E=midpoint(A--C), F=midpoint(A--B), P=centroid(A,B,C); draw(D--A--B--C--A^^B--E^^C--F); dot(A^^B^^C^^P); label("aa", P--A, dir(-90)*dir(P--A)); label("bb", P--B, dir(90)*dir(P--B)); label("cc", P--C, dir(90)*dir(P--C)); label("dd", P--D, dir(90)*dir(P--D)); label("dd", P--E, dir(-90)*dir(P--E)); label("dd", P--F, dir(-90)*dir(P--F)); label("AA", A, SW); label("BB", B, SE); label("CC", C, N); label("PP", P, 1.8*dir(285));[/asy]
geometryratioEuler