Question 26
Let $F$, $F_A$, $F_B$ and $F_C$ be the Feuerbach points of non-equilateral triangle $ABC$, and point $I$ be its incenter. Prove that lines $AF_A$, $BF_B$, $CF_C$ and $IF$ pass through one point. Feuerbach points are the tangent points of the nine-point circle with incircle (touches internally) and three excircles (touch externally) of the triangle: $F$ — tangent point to incircle, $F_A$ — tangent point to excircle that touches $BC$, $F_B$ — tangent point to excircle that touches $AC$, $F_C$ — tangent point to excircle that touches $AB$.
The nine-point circle is the circle that passes through the midpoint of each side of the triangle.
The nine-point circle is the circle that passes through the midpoint of each side of the triangle.