Special line segments of a triangle and their corresponding
intersection points
This document gives a quick overview about what each of the special line
segments of a triangle are and properties of these special line segments and
their corresponding intersection points.
(I) Median
-- A line segment that connects a vertex to the midpoint of the opposite side
-- There are 3 medians in a triangle. These 3 line segments are concurrent.
-- The point where the 3 medians intersect is called the centroid of the circle.
This is the center of gravity of the triangle (balancing point of triangle)
(II) Angle bisector
-- A line segment that bisects an angle of a triangle (cuts angle in half)
-- There are 3 angles bisectors in a triangle. These 3 line segments are
concurrent.
-- The point where the 3 angle bisectors intersect is called the incenter of the
triangle. This point is equidistant from each of the 3 sides of the triangle.
-- You can draw a circle so that the circle touches each side of the sides of
the triangle and the incenter is the center of the circle, and the circle will be
inside the triangle. This circle is inscribed in the triangle.
(III) Perpendicular bisector
-- A line that goes through the midpoint of a side and is perpendicular to that
side.
-- There are 3 perpendicular bisectors in a triangle. These 3 lines are
concurrent.
-- The point where the 3 perpendicular bisectors intersect is called the
circumcenter of the triangle. This point is equidistant from each of the 3
vertices of triangle.
-- You can draw a circle where each of the 3 vertices is on the circle and the
circumcenter is the center of the triangle, and the triangle will be inside the
circle. This circle circumscribes the triangle.
(IV) Altitude
-- A perpendicular line segment that connects a vertex to the side opposite
that vertex.
-- There are 3 altitudes in a triangle. These 3 line segments are concurrent.
-- The point where the 3 altitudes intersect is called the orthocenter of the
triangle.
Note: These 4 points are only going to be the same point in the case of an equilateral
triangle. However, no matter what type of triangle you have, 3 of the points (the
centroid, the circumcenter, and the orthocenter) will always be collinear. The line drawn
through these 3 points is called the Euler line, after Leonard Euler.