Antiparallels

Antiparallels to the base AB of a triangle ABC are called the lines JH, defined by intersecting the sides CA, CB of the triangle with circles passing through AB. The antiparallels to AB are parallel to each other and also parallel to the tangent to the circumcircle passing through C.
A particular antiparallel to side AB is the line [PQ], joining the feet of the altitudes BP, AQ of the triangle.
Thus, antiparallels are parallel to the sides of the orthic triangle.

The middles (M) of the segments JH move on a line CK, which is the [symmedian] of the triangle w.r. to C. The tangent at C and this symmedian make together with the sides of C a [harmonic bundle] of lines, dividing every line intersecting these four lines in [harmonic ratio].
Moving M to the intersection L of the symmedian with the tangent at A, we see that the other tangent, at B, passes also from L. In fact, points M, F the center of the circle {ABD} and L coincide in this case. Hence the Symmedian point of ABC (which is the intersection point of the three symmedians (look at Symmedian.html )) is the Gergonne point of its tangential triangle. Moving M to the position, such that AB = JH (make F and G coincident), we see that the symmedian is the symmetric of the median from C, with respect to the bisector of C.
Look at the file Symmedian_0.html for a more detailed discussion of the symmedian line from a vertex and its properties. File AntiparallelHyperbola.html contains a discussion on the locus of intersection points of lines AH, BJ, which is a rectangular hyperbola passing through the vertices of the triangle.

Antiparallels

See Also