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.

Image link: http://s1.postimg.org/r1p7xmmjj/img.png

Image link: http://s1.postimg.org/r1p7xmmjj/img.png

This problem is adapted from the Proofathon Geometry contest, and was originally posed by Nicolae Shapoval.
×

Problem Loading...

Note Loading...

Set Loading...