Geometry, Geometry theorems
Formulate a complete exposition of triangle geometry as an example of a complete mathematical theory.
- Give a unified algebraic explanation for the centroid, circumcenter, orthocenter, incenter.
Centroid (paths)
- Midpoints of sides connected to vertices.
- Pass through centroid.
- Ratio is 1:2 for each segment through the centroid.
- Center of mass.
Circumcenter (distance)
- Triangle in circle.
- Center is equidistant to all three vertices.
- Obtuse angle: circumcenter is outside of the triangle.
Orthocenter (angles)
- Three altitudes intersect at the orthocenter.
- In the critical case we have a right angle.
- Obtuse anlge - orthocenter is outside of the triangle.
Centroid, circumcenter, orthocenter lie on Euclid's line.
Incenter
- Three angle bisectors meet at the incenter.
- The center of the circle inside the triangle.
- Off of Euclid's line unless the triangle is isosceles.