AoPS - Problem

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Problem

Source: Indonesian TST Stage 1: Test 4 - Geometry

Tags: geometry, circumcircle, concurrency

DavyDuf

26.02.2023 17:23

Given an acute triangle $ABC$ with altitudes $AD$ and $BE$ intersecting at $H$ , $M$ is the midpoint of $AB$ . A nine-point circle of $ABC$ intersects with a circumcircle of $ABH$ on $P$ and $Q$ where $P$ lays on the same side of $A$ (with respect to $CH$ ). Prove that $ED, PH, MQ$ are concurrent on circumcircle $ABC$

CrazyInMath

27.02.2023 07:00

We invert with center

$H$ and radius

$\sqrt{AH\times DH}$ . This would send the nine-point circle to

$\odot(ABC)$ . Let

$X$ be sent to

$X'$ where

$X$ is any point. We have

$A',B',P'$ colinear and

$D,E,P'$ colinear. So

$DE$ and

$PH$ would meet on

$\odot(ABC)$ . Now let the point of concurrency be

$T$ , it suffice to show that

$T,M,Q$ are colinear. We invert with center

$M$ and radius

$AM$ to send the nine point circle to

$DE$ and

$\odot(AHB)$ to

$\odot(ABC)$ Now it follows that the intersection of

$DE$ and

$\odot(ABC)$ would lies on

$QM$ , the conclusion follows.