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21256A conic centered at Euler line

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  • Francisco Javier
    Oct 26, 2012
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      I found the following:

      Theorem. Given a triangle ABC, call Γ the locus of points P such that polar of P with respect to the circumcircle is tangent to the nine point circle. Then we have:
      1) Γ is a conic whose center is X26, the circumcenter of the tangential triangle.
      2) Γ is an ellipse, parabola o hyperbola if and only if the triangle is acute, triangle or obtuse.
      3) The diameter on the transverse axis of the conic is also a diameter of the circumcircle of tangential triangle. Therefore, the transverse axis of the conic is the Euler line of the triangle.
      4) If α, β are the lengths of the transverse and conjugate axes of the conic respectively, we have the relation

      (αβ)^2=1−OH^2/R^2

      5) The foci of the conic are O and O′, where O′ is the reflection of O on the center X26.
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