Mixtilinears and Symmedians

Question

(this is from EGMO)

Prove that angles ATK and LTI are equal.

The hint in the book was about symmedians. I am not sure how to prove that line segment AT is the T symmedian, angle chasing did not work, I tried proving $TM_c * M_cA = AM_b*TM_b$

Answer 1

$CLIT$ is cyclic as proven in the previous exercise hence $\angle LTI = \angle ICL = \angle M_{C}CA = \angle ATM_{C} = \angle ATK$.

Or another solution from the hint, we have that $TA$ is the symmedian of $\triangle TKL$, hence the result.