Problem

Source: Iran second round 2016, day 2, problem 5

Tags: geometry



$ABCD$ is a quadrilateral such that $\angle ACB=\angle ACD$. $T$ is inside of $ABCD$ such that $\angle ADC-\angle ATB=\angle BAC$ and $\angle ABC-\angle ATD=\angle CAD$. Prove that $\angle BAT=\angle DAC$.