Locus: Equidistant from Sides of Angle

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A locus is a set of points which satisfies a certain condition.

theorem Locus Theorem 5
The locus in the interior of an angle equidistant from the sides of the angle is the bisector of the angle (exclusive of the vertex).

This theorem asks you to "describe the path formed by all points the same distance from the sides of an angle".
ANSWER: the angle bisector.
Example 1: A historical society is thinking about adding additional "spoke" supports to this old water wheel. They would like to add an additional spoke between each set of existing spokes such that that the new spoke will be equidistant from the existing spokes. Each new spoke will bisect the angle formed by the existing spokes.

Set of points: new spoke
Condition: equidistant from existing spokes
Locus: angle bisector of angle formed by existing spokes


Example 2: Three friends were hiking in Yellowstone National Park. After getting separated, one hiker arrived in Mammoth, one arrived in Tower Falls, and one hiker is missing. The two friends informed the search party that they were last together at Canyon and they were heading North. They were quite convinced that the missing friend was traveling on a path between the other two. The search party started on a path equidistant from the paths of the first two friends. The path of the search party was the angle bisector of the paths of the first two friends.

Set of points: search party path
Condition: equidistant from 2 paths
Locus: angle bisector of angle formed by the paths of the first two friends



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