Question 135
Let $z_{A}$, $z_{B}$, $z_{C}$ be the affixes (complex coordinates) of the vertices of triangle $ABC$. Let $J$, with affix $z_{J}$, be a point in the interior of triangle $ABC$, and let $S_{a}$, $S_{b}$, $S_{c}$ be the areas of triangles $JBC$, $JAC$, and $JAB$, respectively.
a) Prove that $z_{J} = \frac{S_{a} z_{A} + S_{b} z_{B} + S_{c} z_{C}}{S}$, where $S$ is the area of triangle $ABC$.
b) Prove that $J$ is the incenter of triangle $ABC$ if and only if $\left(2 S_{a} - a r\right) z_{A} + \left(2 S_{b} - b r\right) z_{B} + \left(2 S_{c} - c r\right) z_{C} = 0$, where $r$ is the inradius of triangle $ABC$, and $a, b, c$ are the lengths of the sides $[BC], [AC]$, and $[AB]$, respectively.
a) Prove that $z_{J} = \frac{S_{a} z_{A} + S_{b} z_{B} + S_{c} z_{C}}{S}$, where $S$ is the area of triangle $ABC$.
b) Prove that $J$ is the incenter of triangle $ABC$ if and only if $\left(2 S_{a} - a r\right) z_{A} + \left(2 S_{b} - b r\right) z_{B} + \left(2 S_{c} - c r\right) z_{C} = 0$, where $r$ is the inradius of triangle $ABC$, and $a, b, c$ are the lengths of the sides $[BC], [AC]$, and $[AB]$, respectively.