Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be an acute-angled triangle. The lines
![L_{A}](/media/m/e/1/b/e1b0164e4d7212c7c3006023f2d89832.png)
,
![L_{B}](/media/m/6/0/9/6096a393c2e219229f801694a463ae3c.png)
and
![L_{C}](/media/m/5/4/6/5460c043029b250d19584380965ffb48.png)
are constructed through the vertices
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
,
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
and
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
respectively according the following prescription: Let
![H](/media/m/4/c/0/4c0872a89da410a25f00b86366efece7.png)
be the foot of the altitude drawn from the vertex
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
to the side
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
; let
![S_{A}](/media/m/e/0/1/e01682a073cc75226d74b513a7c391b7.png)
be the circle with diameter
![AH](/media/m/1/7/0/1700da59d12a188862b9dc234aba8941.png)
; let
![S_{A}](/media/m/e/0/1/e01682a073cc75226d74b513a7c391b7.png)
meet the sides
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
and
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
at
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
and
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
respectively, where
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
and
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
are distinct from
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
; then let
![L_{A}](/media/m/e/1/b/e1b0164e4d7212c7c3006023f2d89832.png)
be the line through
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
perpendicular to
![MN](/media/m/2/6/7/267a73297a5de9e529d41774ee6ff45a.png)
. The lines
![L_{B}](/media/m/6/0/9/6096a393c2e219229f801694a463ae3c.png)
and
![L_{C}](/media/m/5/4/6/5460c043029b250d19584380965ffb48.png)
are constructed similarly. Prove that the lines
![L_{A}](/media/m/e/1/b/e1b0164e4d7212c7c3006023f2d89832.png)
,
![L_{B}](/media/m/6/0/9/6096a393c2e219229f801694a463ae3c.png)
and
![L_{C}](/media/m/5/4/6/5460c043029b250d19584380965ffb48.png)
are concurrent.
%V0
Let $ABC$ be an acute-angled triangle. The lines $L_{A}$, $L_{B}$ and $L_{C}$ are constructed through the vertices $A$, $B$ and $C$ respectively according the following prescription: Let $H$ be the foot of the altitude drawn from the vertex $A$ to the side $BC$; let $S_{A}$ be the circle with diameter $AH$; let $S_{A}$ meet the sides $AB$ and $AC$ at $M$ and $N$ respectively, where $M$ and $N$ are distinct from $A$; then let $L_{A}$ be the line through $A$ perpendicular to $MN$. The lines $L_{B}$ and $L_{C}$ are constructed similarly. Prove that the lines $L_{A}$, $L_{B}$ and $L_{C}$ are concurrent.