From the property of the Euler Line (a line which passes through the circumcenter, the orthocenter and the centroid of a triangle) we can build the following structure:
1-The euclidean property is the existence of the Euler Line.
2-Several applets (like this one) have been already created by different authors to show this property.
3-GeoGebra applet for the corresponding stEulerLine.
4-We see that the stEuler Line is the line which passes through the stCircumcenter (the point where the stPerpendiculars of the sides at their centers meet) , the stOrthocenter (the point where the stPerpendiculars of the sides from the opposite vertexes meet) and the centroid of a triangle.
5- Euclidean translation of this property: Reflecting the sides about diagonal directions at their centers they meet at a common point O, reflecting the sides about diagonal directions at their opposite vertexes they meet at a common point C, and joining the vertexes with the centers of the opposite sides we obtain a point B.
Then the points O, C and B are joined by a common line.
6-Substituting the reflections about the diagonal by "directed reflections"
Reflecting the sides about arbitrary parallel directions (directed reflections) at their centers they meet at a common point O, reflecting the sides about the same directions at their opposite vertexes they meet at a commmon point C, and joining the vertexes with the centers of the opposite sides we obtain a point B.
Then the points O, C and B are joined by a common line.
7-We could even realize that the distance between the centroid and the reflOrthocenter is double of the distance between the centroid and the stCircumcenter.
8- Creating the locus for the stCircumcenter as well as for the stOrthocenter, in both cases using as slider a point around a circle to define the reflecting direction, we obtain two circles, one which passes through the centers of the three sides and the other one coincident with the orthocenter.
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, while the directed reflections of the sides of the same triangle at their opposite vertexes coincide at a point, and the locus of these points is the circumcenter of the original triangle. The line which passes through both points passes also through the triangle centroid.
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
Pink lines: Reflections of the sides about the green lines at their centers
Pink point: common point (reflCircumcenter)
Pink circle: Locus of the reflCircumcenters for all the possible reflecting directions: it passes through the centers of the three sides.
Blue lines: Reflections of the sides about the green lines at their opposite vertexes
Blue point: common point (reflOrthocenter)
Blue circle: Locus of the reflOrthocenters for all the possible reflecting directions: It coincides with the circumcenter
Yellow point: Centroid (where the three medians meet)
Red segment: It joins the pink, blue and yellow points.