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Rule

Incenter Theorem

The incenter of a triangle is the point which is equidistant from each of the triangle's sides. This point is considered to be the center of the triangle.

Based on the diagram, the following relation holds true.

$D I = E I = F I$

Proof

Consider a triangle and its incenter $I .$

Let $D I,$ $E I,$ and $F I$ be the distances from $I$ to the sides of the triangle. Recall that the distance from a point to a segment is perpendicular to the segment.

By the definition of an incenter,

\overline{A I}

is the angle bisector of

∠ B A C .

Since

I

lies on

\overline{A I},

it is equidistant from the angle's sides by the Angle Bisector Theorem.

D I = E I

Similarly, since

I

lies on

\overline{B I},

which is the bisector of

∠ A B C,

it is also equidistant from this angle's sides.

E I = F I

By bringing together the above information, the following is obtained.

D I = E I and E I = F I ⇕ D I = E I = F I

This means that

I

is equidistant from each of the triangle's sides.

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Proof