Triangle centers
Three lines drawn in a triangle almost never pass through one point. Why do these four families always do, and what is special about each meeting point?
Where you are 4 inside
What this area is about
Every triangle has four families of special lines: perpendicular bisectors, angle bisectors, medians and altitude lines. In each family the three lines are concurrent, meeting at one point, and that is a theorem, not a lucky drawing. Each meeting point is a center with a job of its own.
How these fit together
All four are concepts, each a point of concurrency plus what that point does.
Centers from equal distances. The circumcenter O, where the perpendicular bisectors meet, is equally far from all three vertices and centers the circle through them; it is inside an acute triangle, at the midpoint of the hypotenuse of a right one, outside an obtuse one. The incenter I, where the angle bisectors meet, is equally far from all three sides and centers the circle inside tangent to them.
Centers from segments. The centroid G, where the medians meet, lies two thirds of the way along each median from its vertex and is the balance point. The orthocenter and the Euler line: the altitude lines meet at H, outside the triangle when it is obtuse, and in every triangle that is not equilateral, O, G and H are three different points on one line.
A path through it
The circumcenter first. The incenter and the centroid both follow it, and the orthocenter comes after the centroid.
After this, inscribed polygons and cyclic quadrilaterals build on the circumcircle, and tangent properties on the incircle.