Let ABC be an acute triangle with circumcircle$$ Γ$$Let$$l$$be a tangent line to$$Γ$$and let$$l_a, l_b, l_c$$be the lines obtained by reflecting$$ l$$in the lines BC, CA and AB, respectively. Show that the circumcircle of the triangle determined by the lines $$l_a, l_b, l_c$$ is tangent to the circle$$Γ$$

Question

Let ABC be an acute triangle with circumcircle$$ Γ$$Let$$l$$be a tangent line to$$Γ$$and let$$l_a, l_b, l_c$$be the lines obtained by reflecting$$ l$$in the lines BC, CA and AB, respectively. Show that the circumcircle of the triangle determined by the lines $$l_a, l_b, l_c$$ is tangent to the circle$$Γ$$

anonymous · Answer

I am confused on this one, how is the graph supposed to be?

asnaseer · Answer

I have shown that the statement is always true using GeoGebra but have not been able to prove it mathematically.
I have attached the interactive GeoGebra files for anyone interested in viewing this. once extracted into a folder, just double-click on the html file to view the interactive construction for this problem.