A Special Triangle with a Line Through the Lemoine Point

Problem

A Special Triangle with a Line Through the Lemoine Point

Proof 1

Let $a^2=x,\,b^2=y,\,c^2=z.\,$ We have $x+y=2z.\,$ We know that

$\displaystyle\overrightarrow{AK}=\frac{y\overrightarrow{AB}+z\overrightarrow{AC}}{x+y+z}=\frac{y\overrightarrow{AB}+z\overrightarrow{AC}}{3z}.$

Also, $\displaystyle\overrightarrow{AG}=\frac{\overrightarrow{AB}+\overrightarrow{AC}}{3},\,$ implying $\overrightarrow{GK}=\overrightarrow{AK}-\overrightarrow{AG}=\displaystyle\frac{(y-z)\overrightarrow{AB}}{3z},\,$ so that, indeed, $GK\parallel AB.$

Proof 2

In homogeneous barycentric coordinates, $G=1:1:1\,$ while $K=a^2:b^2:c^2.\,$ The equation of the side line $AB\,$ is $z=0.\,$ The equation of line $GK\,$ is

$\left|\begin{array}{ccc}\,x&y&z\\1&1&1\\a^2&b^2&c^2\end{array}\right|=(c^2-b^2)x+(a^2-c^2)y+(b^2-a^2)z=0.$

Since $c=\displaystyle\frac{a^2+b^2}{2},\,$ the equation of $GK\,$ is

$(a^2-b^2)x+(a^2-b^2)y+(b^2-a^2)z=0.$

The intersection of $GK\,$ with the line at infinity $x+y+z=0\,$ is $1:-1:0.\,$ This point also lies on $z=0,\,$ implying that the two lines intersect on the line at infinity. Hence, they are parallel.

Acknowledgment

Leo Giugiuc has kindly posted at the CutTheKnotMath facebook page the above problem, due to Kadir Altintas, with his solution (Proof 1) and the solution by the problem's author (Proof 2).

 

Symmedian

  1. All about Symmedians
  2. Symmedian and Antiparallel
  3. Symmedian and 2 Antiparallels
  4. Symmedian in a Right Triangle
  5. Nobbs' Points and Gergonne Line
  6. Three Tangents Theorem
  7. A Tangent in Concurrency
  8. Symmedian and the Tangents
  9. Ceva's Theorem
  10. Bride's Chair
  11. Star of David
  12. Concyclic Circumcenters: A Dynamic View
  13. Concyclic Circumcenters: A Sequel
  14. Steiner's Ratio Theorem
  15. Symmedian via Squares and a Circle
  16. Symmedian via Parallel Transversal and Two Circles
  17. Symmedian and the Simson
  18. Characterization of the Symmedian Point with Medians and Orthic Triangle
  19. A Special Triangle with a Line Through the Lemoine Point

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