Going through the official solution of IMO 2016 Problem G2. Full pdf can be found here: https://www.imo-official.org/problems/IMO2016SL.pdf
I have a couple of questions regarding the solution.
Can someone prove or point me to a proof of the statement given at the end of the first paragraph starting with "it is well-known that angle BAT ..."? For something that is well-known the proof of it is surely hard to find.
In the second paragraph quadrilateral $SFTE$ is obviously harmonic. Why? I know the definition, but can't prove it.
At the end of the 2nd paragraph they project $T$ to infinity and say that $X$ is thus projected to $M$. Why? I played with the cross-ratio, but can't get anything close to the result.
1 and 3 are answered by Mindlack in the comments. His/her intuition how to approach 2 has led me to a proof, which I'm leaving here.
Since $AF$ touches $\omega_A$ at $F$, we have $\angle FTS=\angle SFA$, hence $\triangle AFT \sim \triangle AFS$ and $\frac{FS}{FT}=\frac{AF}{AT}$. Similarly, $\frac{SE}{TE}=\frac{AE}{AT}$. Since $AE=AF$, we get $FS\cdot TE=FT\cdot SE$. Which establishes that $SFTE$ is harmonic.