The intersection at one point (called Gergonne point) of the lines from vertices of a triangle to contact points of the inscribed circle can be proved immediately using Ceva’s theorem.
Is there a direct proof that does not pass through Ceva’s formula?
Edit: I am hoping for a metric Euclidean proof using lengths and angles but this should not limit the answers. If you can prove it using algebraic K theory, go ahead.
It is a conclusion of a degenerated case of the Brianchon’s theorem:
For a hexagon $ABCDEF$ with an inscribed conic section the lines $AD$, $BE$, $CF$ coincide in a common point.
Just set $A$, $C$ and $E$ to be vertices of the triangle, and $B$, $D$ and $F$ the points where the incircle touches the triangle. Some immediate conclusion: the Gergonne point could be generalized to ellipse and other conics. Quick search reveals this idea is not new and there is even some literature on it, e.g. this nice pdf and also this page.
I hope this is what you were looking for 😉