Loading ...
Sorry, an error occurred while loading the content.

More Circles

Expand Messages
  • xpolakis@otenet.gr
    Lat ABC be a triangle and (Ab), (Ac) the circles passing through H and touching BC at B, C, respectively. A / / / / / Ab Ac /
    Message 1 of 2 , Aug 1, 2001
    • 0 Attachment
      Lat ABC be a triangle and (Ab), (Ac) the circles passing through H and
      touching BC at B, C, respectively.


      A
      /\
      / \
      / \
      / \
      / \ Ab
      Ac / \
      / H \
      / \
      / A' \
      / \
      B-----------A"-------C


      A' = BAb /\ CAc

      A" = orth. Proj. of A' on BC

      Similarly we define B',C'; B",C".

      The triangles A'B'C', A"B"C" are in perspective with ABC.

      Perspectors in Barycentrics:

      1. A'B'C', ABC:

      (sinAsec^2Acos(B-C) ::)

      2. A"B"C", ABC:

      (sec^2A ::)

      GENERALIZATION:

      Lat ABC be a triangle, P a point and (Ab), (Ac) the circles passing through P
      and touching BC at B, C, respectively.


      A
      /\
      / \
      / \
      / \
      / \ Ab
      Ac / \
      / P \
      / \
      / A' \
      / \
      B-----------A"-------C


      A' = BAb /\ CAc

      A" = orth. Proj. of A' on BC

      Similarly we define B',C'; B",C".

      1. Which is the locus of P such that A'B'C', ABC are perspective?

      2. Which is the locus of P such that A"B"C", ABC are perspective?

      Answers:

      1. The locus is a sextic.

      2. The locus is the entire plane ie for every P the triangles A"B"C", ABC
      are perspective.

      Perspector:

      (L^(-2) : M^(-2) : N^(-2)), where L,M,N are the tripolar distances of P
      (ie AP, BP, CP, resp.) in Barycentrics

      (sinA / (y^2 + z^2 + 2yzcosA) ::) in Normals


      P Perspector in Normals
      ---------------------------------------
      I (tan(A/2) ::)

      G (1/a(m_a)^2 ::) = (1/a(-a^2 + 2(b^2 + c^2)) ::)
      where m_a = A-median

      O (1/a ::) = G

      H (cscAsec^2A ::)

      K (a/(m_a)^2 ::) = (a/(-a^2 + 2(b^2 + c^2)) ::)

      etc



      Hope that my calculations were correct!

      Antreas
    Your message has been successfully submitted and would be delivered to recipients shortly.