Intersections of Bisectors

Date: 9/6/95 at 14:0:50
From: "$info"
Subject: analytic geometry

Explain how to get the incenter and the circumcenter of a triangle. 
Also, how to get the orthocenter of a triangle.

Date: 9/13/95 at 12:56:21
From: Doctor Steve
Subject: Re: analytic geometry

There are a few key lines in any triangle that are worth exploring 
because they generate so many interesting relations.

Try constructing the angle bisectors and then exploring their 
intersection(s?) in relation to the other elements of the triangle.

Try the same with the perpendicular bisectors, and then again 
with the altitudes, and then again with the medians.

I have written out some answers below where it says "SPOILER." 
Try exploring first before reading those.

-Doctor Steve,  The Geometry Forum

*******SPOILER*********

Angle bisectors intersect at the incenter
Perpendicular bisectors at the circumcenter
Altitudes at the orthocenter
Medians at the centroid

Extra credit question:  what use is the centroid?