Apollonius Point

Consider the Excircles $\Gamma_A$, $\Gamma_B$, and $\Gamma_C$ of a Triangle, and the Circle $\Gamma$ internally Tangent to all three. Denote the contact point of $\Gamma$ and $\Gamma_A$ by $A'$, etc. Then the Lines $AA'$, $BB'$, and $CC'$ Concur in this point. It has Triangle Center Function

$$\alpha=\sin^2 A\cos^2[{\textstyle{1\over 2}}(B-C)].$$

References

Kimberling, C. ``Apollonius Point.'' http://cedar.evansville.edu/~ck6/tcenters/recent/apollon.html.

Kimberling, C. ``Central Points and Central Lines in the Plane of a Triangle.'' Math. Mag. 67, 163-187, 1994.

Kimberling, C.; Iwata, S.; and Hidetosi, F. ``Problem 1091 and Solution.'' Crux Math. 13, 128-129 and 217-218, 1987.