Bariz - Problem

Source: https://artofproblemsolving.com/community/c6h1958463p13536764

Tags: geometry, orthocenter, incenter

parmenides51

21.03.2020 15:46

Let $I$ be the incenter of a scalene triangle $\vartriangle ABC$. In other words, $\overline{AB},\overline{BC},\overline{CA}$ are distinct. Prove that if $D,E$ are two points on rays $\overrightarrow{BA},\overrightarrow{CA}$, satisfying $\overline{BD}=\overline{CA},\overline{CE}=\overline{BA}$ then line $DE$ pass through the orthocenter of $\vartriangle BIC$.

Let $M$ be the midpoint of $BC$ , and $A'$ be the symmetrical point of $A$ against $M$. because $BA' = CA = BD , AD = AB - AC = AE , A'B \parallel AE$ , $(A', D, E)$ is on one line. Also, $AD = AE \Rightarrow$ line $A'DE \parallel $ (angle bisector of $\angle BAC$) $\parallel $ (angle bisector of $\angle BA'C$). So, we have to show that $H$ , orthocenter of $\triangle BIC$ , is on the angle bisector of $\angle BA'C$. Let the incenter of $\triangle BA'C$ as $I'$ $\Rightarrow$ $HB \bot CI \parallel BI' , HC \bot BI \parallel CI'$ $\Rightarrow H $is the excenter of $\triangle BA'C$ , and we are done.