Super beautiful fact

[ins-]

It is given triangle $ABC$. Externally for the triangle with side $BC$ is constructed square with center $A_1$. Externally for the triangle with side $CA$ is constructed regular pentagon with center $B_1$. Externally for the triangle with side $AB$ is constructed regular hexagon with center $C_1$. $l$ is a line from $A$ perpendicular to $B_1C_1$. $m$ is a line from $B$ perpendicular to $C_1A_1$. $n$ is a line from $C$ perpendicular to $A_1B_1$. Prove that $l$, $m$ and $n$ intersects at a common point.

[nnosipov]

Действительно, пересекаются в одной точке.

[ins-]

It is not what I wanted to (re)discover. Because it is not a specific property of the reqular n-gons mentioned.

[nnosipov]

Я полагаю, что этот факт скорее относится к алгебре, чем к геометрии.

[ins-]

It can be proved using symetrals and radical axes properties. It depends of the approach used to prove it.

[nnosipov]

ins- в сообщении #633609 писал(а):

It can be proved using symetrals and radical axes properties.

А где можно посмотреть доказательство?

[ins-]

http://www.artofproblemsolving.com/Foru ... 8&t=503135

I think it is a beautiful looking but stupid problem.

[nnosipov]

ins- в сообщении #633612 писал(а):

but stupid problem

Да уж ... Предупреждать надо. Я уж было подумал о каком-то хитром соотношении между корнями из единицы.

[ins-]

This approach can be used to prove the following fact.

http://www.artofproblemsolving.com/Foru ... 7b#p210283