From the orthocentre property (a point where the three perpendiculars to the sides at the vertexes join) we can build the following structure:
1-The euclidean property is the existence of the orthocenter.
2-Several applets (like this one) have been already created by different authors to show this property.
3-GeoGebra applet for the corresponding stOrthocenter.
4-We see that the stOrthocenter is the point where the three altitudes (perpendiculars to the sides passing through the opposite vertexes) coincide.
5- Euclidean translation of this property: Placing parallel lines for every side at their opposite vertexes, and reflecting them at diagonal directions, we obtain three lines which coincide at one point.
6-Substituting the reflections about the diagonal by "directed reflections" : Placing parallel lines for every side at their opposite vertexes, and reflecting them at arbitrary parallel directions (directed reflections), we obtain three lines which coincide at one point.
7-We could try to construct the "orthic triangle" whose inscribed stCircle is centered at the stOrthocenter, 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 vertexes.
9-We can expresss this property as follows: The directed reflections of the sides of a triangle at their opposite vertexes coincide at a point, and the locus of these points is the circumcenter of the original triangle.
It should be proved analytically however, before calling it a "theorem".
Green point: slider for the arbitrary direction
Green lines: The three reflecting parallel lines at the vertexes
Red lines: reflection of the parallels to the sides about the opposite vertexes
Red point: common point
Red circle: Locus for the common point: it coincides with the triangle circumcenter