The existence of L

The proof that DC are parallel to a fixed direction follows by a remark made in OrthoRectangular.html . There we saw that considering the triangle ABC, having its vertices on the rectangular hyperbola, the fourth intersection point of the circumcircle with the hyperbola is the symmetric to the center of the hyperbola of the orthocenter H of ABC, which lies also on the hyperbola. Thus, joining the middles, we see that GE is parallel to CH, hence orthogonal to AB hence is a fixed line through E (this is L

In the aforementioned reference we noticed also that the angle of two chords of the rectangular hyperbola equals the angle of the lines joining their middles. This implies also that the angle of lines {AB,DC} is fixed, hence the parallelity of DC to a fixed direction (the conjugate of EG).

PowerGeneral.html

Produced with EucliDraw© |