MathLabs

Problem 5

The diagonals ACAC and CECE of the regular hexagon ABCDEFABCDEF are divided by interior points MM and NN respectively, so that AMAC=CNCE=r\frac{AM}{AC}=\frac{CN}{CE}=r. Determine rr if B,M,NB,M,N are collinear.
Step 4 of 4: Choose the admissible root
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In plain words

The geometry selects the positive root.

0<r<1⟹r=330<r<1\quad\Longrightarrow\quad r=\frac{\sqrt3}{3}
Detailed analysis

The point MM is interior, so 0<r<10<r<1. Therefore the positive root of 3r2=13r^2=1 is the required value r=3/3r=\sqrt3/3.