Problem 5

The diagonals

AC

and

CE

of the regular hexagon

ABCDEF

are divided by interior points

M

and

N

respectively, so that

\frac{AM}{AC}=\frac{CN}{CE}=r

. Determine

r

B,M,N

are collinear.

Step 1 of 4: Set the hexagon geometry

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In plain words

Regular-hexagon symmetry supplies the fixed ratios needed by Menelaus.

AC=CE=\sqrt3\cdot AB,\quad X=AC\cap BE,\quad AX=XC=\frac12AC

Detailed analysis

Take the side length as $1$ . The long diagonals $AC$ and $CE$ have length $\sqrt3$ , and symmetry gives that $X=AC\cap BE$ is the midpoint of $AC$ ; also $EB=2$ and $BX=1/2$ .