Geometry Chapter 5.3 Medians and Altitudes of a Triangle
I) Vocabulary:
A) Median of a triangle - is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side.
B) Centroid of the triangle - the point of concurrency of the three medians of a triangle .
C) Altitude of the triangle - is the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side. An altitude can lie inside, on, or outside the triangle.
D) Orthocenter of the triangle - the point of concurrency of the three altitudes of a triangle.
II) Theorems:
A) Concurrency of Medians of a Triangle - the medians of a triangle intersect at a point that is called the centroid and that is two thirds of the distance from each vertex to the midpoint of the opposite side.
Example: If point P is the centroid of ΔABC, then AP = 2/3 AD, BP = 2/3 BF, and CP = 2/3 CE.
B) Concurrency of Altitudes of a Triangle - the lines containing the altitudes of a triangle are concurrent at the orthocenter.
Example: If AE, BF, and CD are the altitudes of ΔABC, then the lines AE, BF and CD intersect at the orthocenter point H.
