Let
![A_1](/media/m/5/a/6/5a6ce1347567551c02239ff8d4ebee67.png)
be the center of the square inscribed in acute triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
with two vertices of the square on side
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
. Thus one of the two remaining vertices of the square is on side
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
and the other is on
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
. Points
![B_1,\ C_1](/media/m/c/7/7/c77861ea0e16fcea11ec821c7b2eb666.png)
are defined in a similar way for inscribed squares with two vertices on sides
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
and
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
, respectively. Prove that lines
![AA_1,\ BB_1,\ CC_1](/media/m/5/5/6/556f737104137ecf070af64ec696423a.png)
are concurrent.
%V0
Let $A_1$ be the center of the square inscribed in acute triangle $ABC$ with two vertices of the square on side $BC$. Thus one of the two remaining vertices of the square is on side $AB$ and the other is on $AC$. Points $B_1,\ C_1$ are defined in a similar way for inscribed squares with two vertices on sides $AC$ and $AB$, respectively. Prove that lines $AA_1,\ BB_1,\ CC_1$ are concurrent.