In the same way we can take the nine-point circle as the starting point to establish a new property:
There is always an equilateral hyperbola passing through the following nine points for every triangle:
-The centers of the three sides.
-The points in the middle of the vertexes and the "reflOrthocenter". This special point is defined, for every arbitrary direction, as follows: For every side, reflect a parallel to it passing at the opposite vertex about the chosen direction (reflAltitude) . The three lines meet at this common point
-The feet of these three lines (reflAltitudes) at the corresponding sides
The center of this hyperbola has the following additional property:
All the centers lie on a common circunference. This circumference passes through
The applet can be seen here:
Green Point: Direction Slider
Green Lines: Parallel Lines to the direction marked by the Slider.
Red Lines: reflAltitudes (reflection of parallels to the sides at the opposite vertexes about the chosen direction)
Red Point: reflOrthocenter (common point for the reflAltitudes)
Pink Points: Feet of the reflAltitudes to the side lines.
Dark Blue Points: Side Centers
Purple Points: Midpoints between the vertexes and the stOrthocenter.
Red Curve: Nine-Point Hyperbola (joining the nine previous points)
Light Blue Point: Center of the Nine-Point equilateral hyperbola
Purple Circle: Locus for the hyperbola centers
Drag any of the triangle vertexes or the direction slider to change the figure.