The following theorems present important properties of triangle components:
Theorem: The medians of a triangle are concurrent.
(Figure 1.a)
Theorem: The lines containing the altitudes of a triangle are concurrent.
(Figure 1.b)
Theorem: The perpendicular bisectors of a triangle are concurrent . (Figure 1.c)
Theorem: The angle bisectors of a triangle are concurrent. (Figure 1.d)
H
G
(a) (b)
P
O
(c) (d)
Figure 1
The angle bisectors and the medians of a triangle always meet in the interior of the triangle. The altitudes and perpendicular bisectors can meet in the exterior or on the triangle itself.
H
(a) (b)
H

O
O
(d) (c)
Figure 2. (a-b) Altitudes that meet outside or on the triangle , (c-d) Perpendicular bisectors that meet outside or on the triangle.
We further define more terms in relationship with triangles.
Centroid (G): The point of intersection of the medians in a triangle.
Orthocenter (H): The point of intersection of the altitudes in a triangle.
Circumcenter (O): The point of intersection of the perpendicular bisectors.
Incenter (P): The point of intersection of the angle bisectors in a triangle.
Some Properties of Triangles
The circumcenter O is the center of a circle called the Circumscribed Circle that contains all three vertices of a triangle.
The incenter P is the center of a circle called the Inscribed Circle in a triangle that meets each side of the triangle in only one point that is not a vertex (the circle is tangent to each side).
P
O
Figure 3

We also are able to establish the following theorem, which exhibits a relationship among the centroid, orthocenter, and circumcenter of a triangle.
Theorem: The centroid, orthocenter, and circumcenter of a triangle are collinear and G lies 2/3 of the distance from H to O.
The line that contains these three points is called the Euler Line, in memory of the Swiss mathematician Leonhard Euler (1707—1763 A.D.) who discovered this relationship.
H
G
O
Figure 4