Download When three or more lines intersect in one point, they are concurrent

D. Legg-Battles
Geometry
5.3 Bisectors in Triangles and 5-4 Medians and Altitudes
NOTES
When three or more lines intersect in one point, they are concurrent. The
point at which they intersect is the point of concurrency.
The perpendicular bisectors of the sides of a triangle are concurrent at a
point equidistant from the vertices.
This point is called the circumcenter and is the center of a circumscribed
circle. (See pp. 301-302)
The bisectors of the angles of a triangle are concurrent at a point
equidistant from the sides of the triangle.
This point is called the incenter and is the center of an inscribed circle.
(See p. 303)
A median of a triangle is a segment whose endpoints are a vertex and the
midpoint of the opposite side.
The medians of a triangle are concurrent at a point that is two thirds the
distance from each vertex to the midpoint of the opposite side. (see p. 309)
The medians are concurrent at a point called the centroid. The centroid is
the center of gravity of the triangle.
An altitude of a triangle is the perpendicular segment from a vertex to the
line containing the opposite side. Unlike angle bisectors and medians, an
altitude of a triangle can be a side of a triangle or it may lie outside the
triangle.
The lines that contain the altitudes of a triangle are concurrent.
They are concurrent at a point called the orthocenter. Note when the
orthocenter is inside, on, or outside the triangle. (See top of p. 311)
These notes are only a summary. You should study your textbook, paying
special attention to the diagrams. Also, see the website for “Concurrent
Lines in Triangles” GSP. It is colored-coded to show all of these types of
concurrent lines.