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12099Lemoine`s conics

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  • Ma. de la Paz Alvarez
    Feb 3, 2006
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      Dear all,

      this is the first time I gather courage to write. Even if I've been working for a long time with synthetic geometry the reason is that I don't work with coordinates, so ETC and many of your messages are too much for me (Sorry, but I havn't havd the time!)

      The thing is that I've been exploring Lemoine with Cabri II plus and have found things for which I do not have an explanation nor have I been able to find references:

      One thing is that the locus of the intersecion of two symmedians with the Lemoine´s axis is an hyperbola that passes through the two other vertex as the one not involved moves in the circumcircle of the triangle.

      The other one is the envelope of the Lemine´s axis as one vertex moves in the same way (which by the way can be an ellipse, parabola or hyperbola depending on the angles of the triangle at the vertex considered)

      Then that the locus of the symmedian point as each vertx moves in a line to the paralel to the other side of the triangle is also an ellipse

      Then I saw that the circumcenter of the tangential triangle is on the Euler line of the original triangle and that the locus of the intersections of the symmedian with the Euler line of one vertex as it moves in the same way, are three cirles tangent to the sides of the original triangle.

      And still I can`t explain this!

      So anyhelp will be very very welcome

      Best regards,
      Ma. de la Paz

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