CircumConics

This set of pages tries to unfold the relations among the different theorems which can be derived from the common -euclidean- "Circumcenter" Theorem.

The three mediatrices (perpendicular lines at the side midpoints) of a triangle meet in a common point.

This point is called "circumcenter" due to its additional property of being the center of a circumference which passes through the three vertexes.

The theorem is presented in this page in two versions: The common Euclidean one (CircumCircle)

and the equivalent Minkopwskian version: CircumHyperbola

CircumCircle

Instead of constructing the mediatrices using the (direct) "Perpendicular through center" Tool, in this case the lines through the midpoints are constructed (small unit circle to the left) making two reflections with an angle of 45º among them:

A vertical direction (light blue) for the first reflection, and

A diagonal direction (dashed line) for the second reflection.

Generalization: CircumEllypses

CircumHyperbola

This construction follows the same structure as the Euclidean case, but making only one reflection about the diagonal (dashed) line.

The resulting common point is the center of a straight equilateral hyperbola (which is the equivalent of a Circle in the Minkowskian plane).

It is interesting to see that the hyperbola´s assymptotes (dashed blue lines) correspond also with the reflecting direction and its perpendicular.

Generalization: CircumHyperbolas

General CircumConics