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.
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.
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.