Problem - 3041
A circle inscribed in $\triangle{ABC}$ (the incircle) is tangent to $BC$ at $X$, to $AC$ at $Y$ , to $AB$ at $Z$. Show that $AX$, $BY$, and $CZ$ are concurrent.
First, we note that $AY = AZ$ because these are the two tangent lines from $A$ to the incircle. Similarly, $BX = BZ$ and $CX = CY$. Then, we have
$$\frac{AZ}{ZB}\cdot\frac{BX}{XC}\cdot\frac{CY}{YA}=1$$
The Ceva's theorem, these three lines are concurrent.