Existence of the Incenter: a Second Look
The three angle bisectors of a triangle meet at incenter of the triangle. Reversing the problem we may ask a relevant question:
Given three concurrent lines: α, β, and γ. Is there always a triangle with the three lines as the angle bisectors. If so, construct the triangle.
Given three concurrent lines: α, β, and γ. Is there always a triangle with the three lines as the angle bisectors. If so, construct the triangle.
Solution
The answer to the question is in positive, and the triangle can be constructed is as follows.
First construct a triangle for which the three lines serve as the altitudes. The orthic triangle of the latter is the one we are after, because of the mirror property the orthic triangles possess.
Obvioiusly, the construction is not unique, what is unique is the shape of the triangles - they all are similar.