Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be an acute triangle. Denote by
![B_0](/media/m/5/a/7/5a7b148f9ae7eef70595a0deebfddd3a.png)
and
![C_0](/media/m/e/9/e/e9eb7207b3e27429b1d887f6793224be.png)
the feet of the altitudes from vertices
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
and
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
, respectively. Let
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
be a point inside the triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
such that the line
![BX](/media/m/3/3/7/337e22dcd76e2ed929a1109e8f7c9176.png)
is tangent to the circumcircle of the triangle
![AXC_0](/media/m/1/5/2/15201ab3893177c88c4e97a6bb409d74.png)
and the line
![CX](/media/m/3/e/9/3e92a8fed18c9f151c4930b9f011439c.png)
is tangent to the circumcircle of the triangle
![AXB_0](/media/m/9/d/4/9d4f77d0f1050f5fb2deff312034b1e9.png)
. Show that the line
![AX](/media/m/3/a/8/3a8b3cfe621304b5621fb712075419c2.png)
is perpendicular to
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
.
%V0
Let $ABC$ be an acute triangle. Denote by $B_0$ and $C_0$ the feet of the altitudes from vertices $B$ and $C$, respectively. Let $X$ be a point inside the triangle $ABC$ such that the line $BX$ is tangent to the circumcircle of the triangle $AXC_0$ and the line $CX$ is tangent to the circumcircle of the triangle $AXB_0$. Show that the line $AX$ is perpendicular to $BC$.