From the circumcenter property (a point where the three mediatrices of the sides meet) we can build the following structure:
1-The euclidean property is the existence of the circumcenter.
2-Several applets (like this one) have been already created by different authors to show this property.
3-GeoGebra applet for the corresponding stCircumcenter.
4-We see that the stCircumcenter is the point where the three mediatrices (perpendiculars to the sides passing through their central points) coincide.
5- Euclidean translation of this property: Reflecting the sides about diagonal directions at their central points, we obtain three lines which coincide at one point.
6-Substituting the reflections about the diagonal by "directed reflections" Reflecting the sides of a triangle at their central points about arbitrary parallel directions (directed reflections), we obtain three lines which coincide at one point.
7-We could try to construct a "circunscribed line" as the curve which passes trough the three vertexes having the "circumcenter" asa central point , but instead we go ahead to the next step:
8- We create the locus for the common point using as slider a point around a circle to define the reflecting direction. This locus is a circle that passes through the three side centers.
9-We can expresss this property as follows: The directed reflections of the sides of a triangle at their centers coincide at a point, and the locus of these points is a circumference passing through the three side centers.
However, this statement should be proved analytically before calling it a "theorem".
Green point: slider for the arbitrary direction
Green lines: The three reflecting parallel lines at the side centers
Red lines: Reflections of the sides about the green lines
Red point: common point
Red circle: Locus for the common point: a circle passing through the side centers