Concurrent Lines, Medians, and Altitude: Geometry Help

Help with concurrent lines, point of concurrency, perpindicular bisectors, angle bisectors, circumcenters of triangles, and incenters of triangles.

Definition: Concurrent Lines

Concurrent Lines are three or more lines intersecting at one point.

Definition: Point of Concurrency

The Point of Concurrency is the point where concurrent lines intersect.

Theorem

Perpendicular bisectors of sides of a triangle are concurrent at a point equidistant from the vertices.

Definition: Circumcenter of Triangle

The Circumcenter of a Triangleis the point of concurrency of the perpendiculars of a triangle.

For example, in the picture below, the three perpendicular bisectors(M¹, M², M³) intersect at point O. That makes point O the circumcenter of the triangle. It is also the center of the circle surrounding the triangle. Therefore, the circle is “circumscribed” about the triangle.

Source

Theorem

Angle bisectors of a triangle are concurrent at a point equidistant from the sides.

Definition: Incenter of Triangle

The Incenter of a Triangle is the point of concurrency of the angle bisectors.

For example, in the picture below, the three angle bisectors(t¹, t², t³) intersect at point I, making point I the incenter and also the center of the circle. Therefore, the circle is inscribed in the triangle.

Source

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One Response to “Concurrent Lines, Medians, and Altitude: Geometry Help”
  1. john Says...

    On October 5, 2010 at 1:17 pm

    thanks it helped out a lot


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