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 3 of 4: Apply Menelaus

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In plain words

Collinearity becomes one scalar equation for the single unknown ratio.

\frac{r}{1-r}\cdot4\cdot\frac{r-1/2}{1-r}=1\quad\Longrightarrow\quad3r^2=1

Detailed analysis

Because $B,M,N$ are collinear, Menelaus in triangle $CBE$ gives the product of the three ratios equal to $1$ . Substituting Step 2 and simplifying yields $4r(r-1/2)=(1-r)^2$ , hence $3r^2=1$ .