Two circles in a parallelogram

Question

Yesterday I was solving geometry problems and I met with following one: Given a parallelogram ABCD and point T on the diagonal AC. Prove that circles through T inscribed into angles BAD and BCD are tangent at T.

I couldn’t solve it, maybe it needs transformations or something. The problem can be found here without the solution (I dont have Java)

https://www.cut-the-knot.org/Curriculum/Geometry/TwoTangentCirclesInPara.shtml#More

Please help to solve it!

Answer 1

Scaling by $-\frac{TC}{TA}$ with centre $T$ maps $\angle BAD$ to $\angle DCB$ and hence the first circle to the second.

Answer 2

Hint:

Denote by $C_1$ and $C_2$ the centers of the circumferences $\omega_1$ and $\omega_2$ respectively. We know that $T\in \omega_1, \omega_2$. If points $C_1, T, C_2\in AC$ (they're collinear), what's the obvious observation?