Let
![H](/media/m/4/c/0/4c0872a89da410a25f00b86366efece7.png)
be the orthocenter of an acute-angled triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
. The circle
![\Gamma_{A}](/media/m/5/d/a/5da953383f4b346be9912f2e7bec0ab2.png)
centered at the midpoint of
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
and passing through
![H](/media/m/4/c/0/4c0872a89da410a25f00b86366efece7.png)
intersects the sideline
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
at points
![A_{1}](/media/m/9/7/4/9742b2655cd943b758073e1f1d090c23.png)
and
![A_{2}](/media/m/a/2/5/a25771e5c2a6c9c6113eab3c46cf63d8.png)
. Similarly, define the points
![B_{1}](/media/m/f/a/5/fa55cf39f6736c287bf64ee9471f00f1.png)
,
![B_{2}](/media/m/5/f/0/5f0ace33ca787cd757fffa052940b6a9.png)
,
![C_{1}](/media/m/e/4/6/e46111370b6102ad343bcdc7190d9ff9.png)
and
![C_{2}](/media/m/1/3/c/13ce248ee56b45a1b9e032736b8100a1.png)
.
Prove that six points
![A_{1}](/media/m/9/7/4/9742b2655cd943b758073e1f1d090c23.png)
,
![A_{2}](/media/m/a/2/5/a25771e5c2a6c9c6113eab3c46cf63d8.png)
,
![B_{1}](/media/m/f/a/5/fa55cf39f6736c287bf64ee9471f00f1.png)
,
![B_{2}](/media/m/5/f/0/5f0ace33ca787cd757fffa052940b6a9.png)
,
![C_{1}](/media/m/e/4/6/e46111370b6102ad343bcdc7190d9ff9.png)
and
![C_{2}](/media/m/1/3/c/13ce248ee56b45a1b9e032736b8100a1.png)
are concyclic.
Author: Andrey Gavrilyuk, Russia
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Let $H$ be the orthocenter of an acute-angled triangle $ABC$. The circle $\Gamma_{A}$ centered at the midpoint of $BC$ and passing through $H$ intersects the sideline $BC$ at points $A_{1}$ and $A_{2}$. Similarly, define the points $B_{1}$, $B_{2}$, $C_{1}$ and $C_{2}$.
Prove that six points $A_{1}$ , $A_{2}$, $B_{1}$, $B_{2}$, $C_{1}$ and $C_{2}$ are concyclic.
Author: Andrey Gavrilyuk, Russia