Given is a triangle $ABC$ and let $D$ be the midpoint the major arc $BAC$ of its circumcircle. Let $M , N$ be the midpoints of $AB , AC$ and $J , E , F$ are the touchpoints of the incircle $(I)$ of $\triangle ABC$ with $BC, CA, AB$. The line $MN$ intersects $JE , JF$ at $K , H$ respectively; $IJ$ intersects the circle $(BIC)$ at $G$ and $DG$ intersects $(BIC)$ at $T$. a) Prove that $JA$ passes through the midpoint of $HK$ and is perpendicular to $IT$. b) Let $R, S$ respectively be the perpendicular projection of $D$ on $AB, AC$. Take the points $P, Q$ on $IF , IE$ respectively such that $KP$ and $HQ$ are both perpendicular to $MN$. Prove that the three lines $MP , NQ$ and $RS$ are concurrent .
Problem
Source: VMO 2023 day 1 P4
Tags: geometry
24.02.2023 20:37
a_507_bc wrote: $JA$ intersects the circle $(BIC)$ at $G$ Is it correct? I think it must be $JA$ intersects the circle $(ABC)$ at $G$.
24.02.2023 20:59
It is IJ, sorry for the typo
24.02.2023 21:00
First part of a) $JA$ passes through the midpoint of $HK$ is just Iran Lemma. We know that $B, I, K$ are collinear and $\angle BKA = 90^\circ$, so $\angle AKH = 90^\circ - \angle HKB = 90^\circ - \frac{1}{2}\angle B = \angle FJB = \angle JHK$, so $AK$ and $HJ$ are parallel. And because $MN$ bisects $AJ$, then $AJ$ bisect $HK$. ($AHJK$ is a parallelogram).
24.02.2023 22:59
25.02.2023 08:21
this geo easy for a VMO's
25.02.2023 10:22
Some good looking properties (May be c)) Prove that: $PQ \perp IZ$ with $Z$ midpoint $BC$ Prove that: $PQ$ bisects $IZ$. (Collected)
25.02.2023 18:40
Way too many points to look at Part a is easy For part b, the lines concurrent at Spieker Center
25.02.2023 18:44
Well, in recent years VMO geometry problems aren't good. Part a and b are independent of each other, and the committee just use old ideas to create these unsatisfying geo
25.02.2023 18:47
wardtnt1234 wrote: Well, in recent years VMO geometry problems aren't good. Part a and b are independent of each other, and the committee just use old ideas to create these unsatisfying geo Yes, you are right. We just need 1 high quality and good problem in the test more than 2 independent problems but not very nice and complicated. There are still more issues about this but I will just say about the Geometry...
10.07.2023 15:49
$\text{Parts a. and b. are solved in separate images because they are not related to each other.}$