**The centroid (G) of a triangle
is the common intersection of the three medians. A median of a
triangle is the segment from a vertex to the midpoint of the opposite
side.**

**The orthocenter (H) of a triangle
is the common intersection of the three lines containing the altitutdes.
An altitude is a perpendicular segment from a vertex to the line
of the opposite side. H does not have to be on the segments that
are the altitudes. In fact, H lies on the lines extended along
the altitudes.**

**The incenter (I) of a triangle
is the point on the interior of the triangle that is equidistant
form the three sides. Since a point interior to an angle that
is equidistant from the two sides of the angle lies on the angle
bisector, then the incenter must be on the angle bisector of each
angle of the triangle.**

**Now we have G, H , C, and I for
the same triangle. I want to now look for relationships among
them and explore these relationships for the many shapes of triangles
(obtuse, right, and acute).**

**First, let's look at an obtuse
triangle.**

**As we notice, the orthocenter
is outside of the triangle.**

**Now let's look at a right triangle.**

**It appears that the orthocenter
goes through the vertex and the right angles and I and G stay
the same distance apart.**

**Now let's look at an acute triangle.**

**it now appears that the orthocenter
has left the interior of the triangle again. This time it exited
through vertex A. I and G still appear to be the same distance
apart.**