Competition problem — Geometry — Maths Olympiad Prep

Library / /8 of 16

Geometry Difficulty 6.4 National Olympiad Prove it JBMO

Problem:

Let $ABC$ be an acute angled triangle with orthocenter $H$ and circumcenter $O$ . Assume the circumcenter $X$ of $BHC$ lies on the circumcircle of $ABC$ . Reflect $O$ across $X$ to obtain $O'$ , and let the lines $XH$ and $O'A$ meet at $K$ . Let $L$ , $M$ and $N$ be the midpoints of $[XB]$ , $[XC]$ and $[BC]$ , respectively. Prove that the points $K$ , $L$ , $M$ and $N$ are cocyclic.

Solution

Solution:

The circumcircles of $ABC$ and $BHC$ have the same radius. So, $XB = XC = XH = XO = r$ (where $r$ is the radius of the circle $ABC$ ) and $O'$ lies on $C(X, r)$ . We conclude that $OX$ is the perpendicular bisector for $[BC]$ . So, $BOX$ and $COX$ are equilateral triangles.

It is known that $AH = 2ON = r$ . So, $AHO'X$ is a parallelogram, and $XK = KH = r/2$ . Finally, $XL = XK = XN = XM = r/2$ . So, $K$ , $L$ , $M$ and $N$ lie on the circle $c(X, r/2)$ .

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.