, , and are three (distinct) non-collinear points in the Cartesian plane, and , , and . The incenter of the triangle is
The x-coordinate of the incenter is a "weighted average" of the x-coordinates of the vertices of the given triangle, and the y-coordinate of the incenter is the same "weighted average" of the y-coordinates of the same vertices. I am requesting an explanation for this statement.
Answer & Explanation
The bisector of angle intersects side at a point , and according to angle bisector theorem we have: . It follows that is a weighted average of and , , with weights given by the lengths of the opposite sides:
and of course we have analogous expressions for the similarly defined points and .
The incenter of is the intersection of , and . It is then hardly surprising that it turns out to be the weighted average of , and . For instance: as belongs to segment we can write:
for some . But the expression for must be symmetric when exchanging , , and among them, and it is easy to verify that does the trick, leading to your formula for :
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