Let
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
be the midpoint of the side
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
of a given triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
. Let
![L](/media/m/f/c/1/fc1ae4eb78da7d1352cbf1f8217ab286.png)
and
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
be points on the sides
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
and
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
, respectively, such that
![\angle CLK = \angle KMC](/media/m/3/3/f/33f578774488c33eb24edc3016a2249b.png)
. Prove that the perpendiculars to the sides
![AB, AC,](/media/m/9/a/7/9a7026254aaa88f69caa4e8a508e6c27.png)
and
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
passing through
![K,L,](/media/m/6/e/0/6e014812f0389a952cb916dcac7cc615.png)
and
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
, respectively, are concurrent.
%V0
Let $K$ be the midpoint of the side $AB$ of a given triangle $ABC$. Let $L$ and $M$ be points on the sides $AC$ and $BC$, respectively, such that $\angle CLK = \angle KMC$. Prove that the perpendiculars to the sides $AB, AC,$ and $BC$ passing through $K,L,$ and $M$, respectively, are concurrent.