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Section 6.6 Concurrence of Lines. A number of lines are concurrent if they have exactly one point in common. m, n and p are concurrent. A. m. n. p. Concurrent lines in Triangles. Theorem 6.6.1: The three angle bisectors of the angles of a triangle are concurrent.

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section 6 6 concurrence of lines

Section 6.6 Concurrence of Lines

A number of lines are concurrent if they have exactly one point in common.

m, n and p are concurrent.

A

m

n

p

Section 6.6 Nack

concurrent lines in triangles
Concurrent lines in Triangles
  • Theorem 6.6.1: The three angle bisectors of the angles of a triangle are concurrent.
  • The point at which the angle bisectors meet is the incenter of the triangle. It is the center of the inscribed circle of the triangle.

Section 6.6 Nack

perpendicular bisectors
Theorem 6.62: The three perpendicular bisectors of the sides of a triangle are concurrent.

The point at which the perpendicular bisectors of the sides of a triangle meet is the circumcenter (center of the circumscribed circle) of the triangle.

Perpendicular Bisectors

Section 6.6 Nack

altitudes of a triangle
Altitudes of a Triangle
  • Theorem 6.63: The three altitudes of a triangle are concurrent.
  • The point of concurrence for the three altitudes of a triangle is the orthocenter of the triangle.

Section 6.6 Nack

medians
Medians

Theorem 6.6.4: The three medians of a triangle are concurrent at a point that is two-thirds the distance from any vertex to the midpoint of the opposite side. The point of concurrence for the three medians is the centroid of the triangle.

Reminder: A median joins a vertex to the midpoint of the opposite side of the triangle.

Section 6.6 Nack

summary
Summary
  • Summary of Chapter Six is on pages 329-330

Section 6.6 Nack