Page 1 of 1

### National Camp 2016

Posted: Mon Feb 20, 2017 4:14 pm
Let ABC be an arbitrary triangle. On the three sides of triangle ABC , three regular n-gons external to the triangle are drawn. Find all values of n for which the centers of the n-gons are vertices of an equilateral triangle.

### Re: National Camp 2016

Posted: Mon Sep 25, 2017 3:48 am
If \$ABC\$ is an arbitrary triangle and the value of \$n\$ satisfies the given condition,obviously it will be also true for certain \$ABC\$.Consider a triangle \$ABC\$ with angle \$\$A=120\$\$, \$\$AB=AC\$\$, circumcenter \$O\$, midpoint of \$\$AB=M\$\$, midpoint of \$\$AC=N\$\$, center of the regular polygon constructed outside of \$AB\$ is \$Z\$ and \$AC\$ is \$Y\$ and \$BC\$ is \$A\$.
for all \$n\$, \$YZ\$ is parallel to \$MN\$ and so triangle \$ZYO\$ is similar to \$MNO\$, which is equilateral. Since \$X\$ belongs to the perpendicular bisector of \$BC\$, \$\$X=O\$\$ and which is possible only then, when \$\$n=3\$\$.
So, \$\$n=3\$\$ can be one solution and it is(\$XYZ\$ is a Nepoleon triangle).So, there exist exactly one value of \$n\$ which is \$3\$.