MathDB

In regular hexagon

ABCDEF

AC

CE

are two diagonals. Points

M

N

are on

AC

CE

respectively and satisfy

AC: AM = CE: CN = r

. Suppose

B, M, N

are collinear, find

100r^2

. [asy] size(120); defaultpen(linewidth(0.7)+fontsize(10)); pair D2(pair P) { dot(P,linewidth(3)); return P; } pair A=dir(0), B=dir(60), C=dir(120), D=dir(180), E=dir(240), F=dir(300), N=(4*E+C)/5,M=intersectionpoints(A--C,B--N)[0]; draw(A--B--C--D--E--F--cycle); draw(A--C--E); draw(B--N); label("

A

",D2(A),plain.E); label("

B

",D2(B),NE); label("

C

",D2(C),NW); label("

D

",D2(D),W); label("

E

",D2(E),SW); label("

F

",D2(F),SE); label("

M

",D2(M),(0,-1.5)); label("

N

",D2(N),SE); [/asy]

Problem Statement