Spring 2001

Essay 1

Concurrency Theorems

The bisectors of the interior angles of any triangle are concurrent at a point (incenter) equidistant from the three sides.

CASE 1: An Acute Triangle

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Let D ABC be an acute triangle. Let l be the line which contains the angle bisector of < BAC. Let m be the angle bisector of < ABC. Let point I be the point of intersection of l and m.

Since I lies on the angle bisector of < BAC, I is equidistant from the sides (AB and AC) of < BAC. (Angle Bisector Theorem) Let IF and IE be perpendicular segments from I to the sides AB and AC, respectively. Then, IF = IE.

Since I lies on the angle bisector of < ABC, I is equidistant from the sides (AB and BC) of < ABC. (Angle Bisector Theorem)

Let IF and ID be perpendicular segments from I to the sides AB and BC, respectively. Then, IF = ID.
Since IE = IF and IF = ID, then IE = ID. (Transitive Property) Therefore, I is equidistant to the sides (BC and AC) of < ACB. I must also lie on the angle bisector of < ACB. (Converse of the Angle Bisector Theorem) Therefore, the three angle bisectors are concurrent at a point I equidistant from the three sides. This point is called the
incenter. The incenter is the center of the inscribed circle of D ABC.

CASE 2: An Obtuse Triangle

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Let D ABC be an obtuse triangle with < ABC as an obtuse angle. All the conditions in the proof for an acute triangle hold for an obtuse triangle.

CASE 3: A Right Triangle

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Let D ABC be a right triangle with < ABC as a right angle. All the conditions in the proof for an acute triangle hold for an obtuse triangle. Additionally, notice that ID || AB since both AB and ID are perpendicular to BC. Also, IF || BC since both BC and IF are perpendicular to AB.

The incenter is always located inside a triangle since it lies within the interior of each of the three angles.

Other concurrency theorems

OTHER ESSAYS