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Parallelian


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.

ParallelianTheorem

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

Delta_A=(a^2alpha^2)/((aalpha+bbeta+cgamma)^2)Delta
(1)
Delta_B=(b^2beta^2)/((aalpha+bbeta+cgamma)^2)Delta
(2)
Delta_C=(c^2gamma^2)/((aalpha+bbeta+cgamma)^2)Delta,
(3)

so it immediately follows that

 sqrt(Delta)=sqrt(Delta_A)+sqrt(Delta_B)+sqrt(Delta_C)
(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,

Delta_(P_(BC)P_(CB)A)=((bbeta+cgamma)^2)/((aalpha+bbeta+cgamma)^2)Delta
(5)
Delta_(P_(CA)P_(AC)B)=((aalpha+cgamma)^2)/((aalpha+bbeta+cgamma)^2)Delta
(6)
Delta_(P_(AB)P_(BA)C)=((aalpha+bbeta)^2)/((aalpha+bbeta+cgamma)^2)Delta,
(7)

giving

 2sqrt(Delta)=sqrt(Delta_(P_(BC)P_(CB)A))+sqrt(Delta_(P_(CA)P_(AC)B))+sqrt(Delta_(P_(AB)P_(BA)C)).
(8)
ParallelianTheorem3

Similarly,

Delta_(PP_(AB)P_(AC))=(bcbetagamma)/((aalpha+bbeta+cgamma)^2)Delta
(9)
Delta_(PP_(BC)P_(BA))=(acalphagamma)/((aalpha+bbeta+cgamma)^2)Delta
(10)
Delta_(PP_(CA)P_(CB))=(abalphabeta)/((aalpha+bbeta+cgamma)^2)Delta,
(11)

so

 Delta_A·Delta_B·Delta_C=Delta_(PP_(AB)P_(AC))·Delta_(PP_(BC)P_(BA))·Delta_(PP_(CA)P_(CB)).
(12)
ParallelianEllipse

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

alpha_0=alpha(a^2alpha^2-abalphabeta-acalphagamma-2bcbetagamma)
(13)
beta_0=beta(-abalphabeta+b^2beta^2-2acalphagamma-bcbetagamma)
(14)
gamma_0=gamma(-2abalphabeta-acalphagamma-bcbetagamma+c^2gamma^2).
(15)

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

ParallelianHomothecy

Consider the anticevian triangle of P and apply the homothecy h(1/2,P) to it. This >triangle is then the triangle formed by the lines (P_(AB), P_(AC)), (P_(BC), P_(BA)), and (P_(CA), P_(CB)) (P. Moses, pers. comm., Nov. 16, 2005).

Let Q be another point in triangle DeltaABC. Let Q_A, Q_B and Q_C be the point Q defined in triangles DeltaPP_(BA)P_(CA), DeltaP_(AB)BP_(CB) and DeltaP_(AC)P_(BC)C respectively, and let Q_A^', Q_B^' and Q_C^' be the point Q defined in triangles DeltaAP_(BC)P_(CB), DeltaP_(AC)BP_(CA) and DeltaP_(AB)P_(BA)C. Triangles DeltaQ_AQ_BQ_C and DeltaQ_A^'Q_B^'Q_C^' are symmetric about the midpoint of segment PQ, the six vertices lie on a central conic. This central conic is a circle if and only if P is the orthocorrespondent of Q (Gibert and van Lamoen 2003).


See also

Equal Parallelians Point, Lemoine Hexagon, Thomsen's Figure, Triangulation Point, Tucker Circles

Portions of this entry contributed by Floor van Lamoen

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References

Gibert, B. and van Lamoen, F. M. "The Parasix Configuration and Orthocorrespondence." Forum Geom. 3, 169-180, 2003. http://forumgeom.fau.edu/FG2003volume3/FG200318index.html.

Referenced on Wolfram|Alpha

Parallelian

Cite this as:

van Lamoen, Floor and Weisstein, Eric W. "Parallelian." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Parallelian.html

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