In triangle $ABC$ with incenter $I$ and circumcircle $\omega$, the tangent through $C$ to $\omega$ intersects $AB$ at point $D$. The angle bisector of $\angle CDB$ intersects $AI$ and $BI$ at $E$ and $F$, respectively. Let $M$ be the midpoint of $[EF]$. Prove that line $MI$ passes through the midpoint of arc $ACB$ of $w$ .
Problem
Source: OLCOMA Costa Rica National Olympiad, Final Round, 2017 Shortlist G4 day2
Tags: geometry, arc midpoint, circumcircle
22.09.2021 15:13
parmenides51 wrote: In triangle $ABC$ with incenter $I$ and circumcircle $w$, the tangent through $C$ to $w$ intersects $AB$ at point $D$. Let $M$ be the midpoint of $AB$. Prove that line $MI$ passes through the midpoint of arc $ACB$ of $w$. There is a typo I guess. We defined $D$ but didn't use it. Also, $MI$ doesn't pass through the midpoint of arc $ACB$ of $w$.
22.09.2021 21:06
This is a bit strange problem, it had defined 3 points at the start not used later, $D$,$E$,$F$ probably to make it easier. I removed all those data. I have the official solution, so indeed $MI$ passes through the arc midpoint of $ACB$. (it has an acute triangle in the solution's figure)
22.09.2021 21:10
$I$-symmedian passes through the midpoint of arc $ACB$.
22.09.2021 21:15
page 5, problem 6, here is the official wording and solution
22.09.2021 21:43
BarisKoyuncu wrote: Also, $MI$ doesn't pass through the midpoint of arc $ACB$ of $w$. indeed, geogebra agrees with what you just said, therefore the above solved problem is wrong, unless I miss something in the translation
23.09.2021 00:22
I think I got what it is trying to say. Here is the problem. In triangle $ABC$ with incenter $I$ and circumcircle $\omega$, the tangent through $C$ to $\omega$ intersects $AB$ at point $D$. The angle bisector of $\angle CDB$ intersects $AI$ and $BI$ at $E$ and $F$, respectively. Let $M$ be the midpoint of $[EF]$. Prove that line $MI$ passes through the midpoint of arc $ACB$ of $w$.
23.09.2021 09:15
23.09.2021 10:04