Let
![AH_1, BH_2, CH_3](/media/m/9/7/0/970ac488ae8628d56b807e1581a6ce3a.png)
be the altitudes of an acute angled triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
. Its incircle touches the sides
![BC, AC](/media/m/d/a/d/dad96fadd6ef89725d9748b99ce63960.png)
and
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
at
![T_1, T_2](/media/m/8/4/3/843ed735a75e2754781ea99e6574ce3e.png)
and
![T_3](/media/m/e/4/6/e4616303b747378d2eb23050a4d12775.png)
respectively. Consider the symmetric images of the lines
![H_1H_2, H_2H_3](/media/m/1/b/e/1be70a355d7e7820870bfdeb0e03fdbe.png)
and
![H_3H_1](/media/m/3/c/3/3c3bb0a73f4fcb574767f2c01360f057.png)
with respect to the lines
![T_1T_2, T_2T_3](/media/m/4/b/7/4b7fba49f38c9b9466492dcc1c7d17fa.png)
and
![T_3T_1](/media/m/a/7/f/a7f62fcb0c0c080cdf26f158b949bc30.png)
. Prove that these images form a triangle whose vertices lie on the incircle of
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
.
%V0
Let $AH_1, BH_2, CH_3$ be the altitudes of an acute angled triangle $ABC$. Its incircle touches the sides $BC, AC$ and $AB$ at $T_1, T_2$ and $T_3$ respectively. Consider the symmetric images of the lines $H_1H_2, H_2H_3$ and $H_3H_1$ with respect to the lines $T_1T_2, T_2T_3$ and $T_3T_1$. Prove that these images form a triangle whose vertices lie on the incircle of $ABC$.