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 2 of 4: Compute the three Menelaus ratios

In plain words

The parameter $r$ appears only in the two ratios on the diagonals.

\frac{CN}{NE}=\frac{r}{1-r},\quad\frac{EB}{BX}=4,\quad\frac{XM}{MC}=\frac{r-1/2}{1-r}

Detailed analysis

The first ratio follows from $CN/CE=r$ . Since $XM=AM-AX$ , the third ratio is $(r-1/2)/(1-r)$ . The middle ratio is $EB/BX=2/(1/2)=4$ .