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This essential trains for: AMC-10, AMC-12, AIME.

An interior angle bisector of a triangle is a ray that bisects an interior angle of the triangle.

Remember that the distance between a point and a line is the length of the perpendicular segment from the point to the line. This segment is unique.

Since any two non-parallel lines in a plane will intersect, two of the bisectors of the interior angles of a triangle will have a common point. Let us denote it with K:

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What information do we have about K?

Since K is on the bisector of the angle ∠CAB, it is equidistant from the sides AB and AC:

equation

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Since K is on the bisector of the angle ∠CBA, it is equidistant from the sides AB and BC:

equation

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By transitivity,

equation

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This means that K is equidistant from the sides CA and CB. Therefore, it is a point on the angle bisector of the side AC.

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The bisectors of the angles of a triangle are concurrent (i.e., they have a common point).

The point of intersection of the angle bisectors is equidistant from all the sides of the triangle. Therefore, it is the center of a circle that is tangent to all three sides - the incenter.

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