Parallelian

A parallelian is a line drawn parallel to one side of a triangle. The three lines drawn through a given point are known as the triangle's parallelians.

There exists a unique point in the interior of a triangle that results in three parallelians of equal length. This point is known as the equal parallelians point.

There is a beautiful theorem connecting the areas of the three triangles determined by parallelians with the area of the reference triangle. Given parallelians through a point with trilinear coordinates , the area of the triangles illustrated above are given by

			(1)
			(2)
			(3)

so it immediately follows that

(4)

(G. Dalakishvili, pers. comm., May 31, 2005). Based on the appearance of the configuration in the theorem, it might be appropriate to term it the "radiation symbol theorem."

Similar theorems also hold for other sets of triangles in the figure (van Lamoen, pers. comm., Dec. 2, 2005). In particular,

			(5)
			(6)
			(7)

giving

(8)

Similarly,

			(9)
			(10)
			(11)

so

(12)

As illustrated above, when lies inside the Steiner inellipse, the endpoints of the parallelians lie on an ellipse with center

			(13)
			(14)
			(15)

If lies on the Steiner inellipse, the points lie on a parabola, and if lies outside the Steiner inellipse, the points lie on a hyperbola. If lies on the Steiner circumellipse, the conic degenerates to straight lines (P. Moses, pers. comm., Nov. 17, 2005).

Consider the anticevian triangle of and apply the homothecy to it. This >triangle is then the triangle formed by the lines (, ), (, ), and (, ) (P. Moses, pers. comm., Nov. 16, 2005).

Let be another point in triangle . Let , and be the point defined in triangles , and respectively, and let , and be the point defined in triangles , and . Triangles and are symmetric about the midpoint of segment , the six vertices lie on a central conic. This central conic is a circle if and only if is the orthocorrespondent of (Gibert and van Lamoen 2003).