Concurrency of perpendiculars through touch points of excircles with sides of a triangle

Question

Let $ABC$ be a triangle. The ex-circles touch sides $BC, CA$ and $AB$ at points $U, V$ and $W$, respectively. Be $r_u$ a straight line that passes through $U$ and is perpendicular to $BC$, $r_v$ the straight line that passes through $V$ and is perpendicular to $AC$ and $r_w$ the straight line that passes through $W$ and is perpendicular to $AB$. Prove that the lines $r_u$, $r_v$ and $r_w$ pass through the same point.

Attempt: I think I saw a property somewhere in this problem: '' circuncenter of $ I_a I_b I_c $ and isogonal conjugates '', but I can't demonstrate it. Is there any way to solve this problem, or to solve it using the mentioned property? I really have no ideas for the problem

Answer 1

You're thinking of correct property. Here are the hints :

• Recall $I$ is orthocenter of $\triangle I_aI_bI_c$ where $A,B,C$ are the feet of altitudes.
• Show that the three $r_i$'s are isogonal lines of those altitudes. For this it is enough to show that $\angle CI_bV=C/2$ and $\angle BI_bI_c=C/2$. As a further hint, use quadrilateral $AIBI_c$ to show the second equality.

Hence the $r_i$'s will concur at isogonal conjugate of the orthocenter $I$, which will be the circumcenter of $\triangle I_aI_bI_c.$