I wonder how I could prove $AC\perp OD$ when the incircle of $\triangle ABC$ has been constructed as shown in the image below without using the argument that $AC$ is a tangent of the circle.
Using the following definition:
the incircle of a triangle is the circle which has exactly one common point with each side of the triangle.
I was trying something like this, but it seems like circular reasoning:
Since $|DO|=|OF|=r$ and $O$ is the interesection point of the 3 angular bisectors. And the angular bisector $AO$ is the set of point which are located at the same distance from $AC$ and $AB$ we can conclude that $|DO|$ is the distance of $O$ to $AC$ which results in perpendicularity.
(however, why does the definition above implies $O$ to be the intersection of the angular bisectors...)