Angle Bisector

The (interior) bisector of an Angle is the Line or Line Segment which cuts it into two equal Angles on the same ``side'' as the Angle.

The length of the bisector of Angle $A_1$ in the above Triangle $\Delta A_1A_2A_3$ is given by

$${t_1}^2=a_2 a_3\left[{1-{{a_1}^2\over(a_2+a_3)^2}}\right],$$

where $t_i\equiv \overline{A_iT_i}$ and $a_i\equiv\overline{A_jA_k}$. The angle bisectors meet at the Incenter $I$, which has Trilinear Coordinates 1:1:1.

See also Angle Bisector Theorem, Cyclic Quadrangle, Exterior Angle Bisector, Isodynamic Points, Orthocentric System, Steiner-Lehmus Theorem, Trisection

References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 9-10, 1967.

Dixon, R. Mathographics. New York: Dover, p. 19, 1991.

Mackay, J. S. ``Properties Concerned with the Angular Bisectors of a Triangle.'' Proc. Edinburgh Math. Soc. 13, 37-102, 1895.