Heptagon (septagon) diagonal intersection

Question

Diagonals $TV$ and $UW$ of regular heptagon $TUVWXYZ$ meet at $A$. Prove that $TU+TA=TW$ (Source: AoPS ItG).

My observations:

• $TUVW$ is an isosceles trapezoid.
• Triangle $TUA$ is congruent to triangle $WVA$.

Answer 1

A hint:

Choose $B$ on $WT$ such that $|WB|=|UT|$, and connect $A$ with $B$. Then find the angles of the triangles $TUA$, $ABW$, and $TAB$. They are all multiples of ${\pi\over7}$. (Note that two of these triangles are congruent.)