# The Concurrent Lines

Geometry Level 4

In acute angled $$\triangle ABC$$, $$\angle ABC = 60^{\circ}$$.

Circles $$\omega_1, \omega_2, \omega_3$$ are constructed having diameters $$BC, CA, AB$$ respectively.

The two tangents from $$A$$ to $$\omega_1$$ touch $$\omega_1$$ at points $$A_2$$ and $$A_3$$, the two tangents from $$B$$ to $$\omega_2$$ touch $$\omega_2$$ at points $$B_2$$ and $$B_3$$, and the two tangents from $$C$$ to $$\omega_3$$ touch $$\omega_3$$ at points $$C_2$$ and $$C_3$$. It turns out that the lines $$A_2A_3, B_2B_3, C_2C_3$$ concur at a point $$T$$ within the triangle. Find $$\angle BAT$$ in degrees.

Details and assumptions

• You might refer to the following figure to see how points $$A_2$$ and $$A_3$$ are constructed. Points $$B_2, B_3, C_2, C_3$$ are defined analogously.