Cevians in a triangle
Source:
March 19, 2006
geometryratioEuler
Problem Statement
Let be an interior point of triangle and extend lines from the vertices through to the opposite sides. Let , , , and denote the lengths of the segments indicated in the figure. Find the product if and .[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("", P--A, dir(-90)*dir(P--A));
label("", P--B, dir(90)*dir(P--B));
label("", P--C, dir(90)*dir(P--C));
label("", P--D, dir(90)*dir(P--D));
label("", P--E, dir(-90)*dir(P--E));
label("", P--F, dir(-90)*dir(P--F));
label("", A, SW);
label("", B, SE);
label("", C, N);
label("", P, 1.8*dir(285));[/asy]