(Advanced Plane Geometry Dao Thanh Oai)

  Let ABC be a triangle, (Oe) be the circle touching to the circumcircle at A and touching the incircle at Ae. Define Be, Ce cyclically. Let A'B'C' be the tangent line of the incircle at Ae, Be, Ce. Then A'B'C' also perspective with many triangle as follows:
  1. A'B'C' perspective with ABC.
    Perspector: X(354)
  2. A'B'C' perspective with excentral triangle.
    Perspector: X(57)
  3. A'B'C' perspective with Cevian triangle of Gergonne point (the perspector is the same point with 2.)
    Perspector: X(57)
  4. A'B'C' perspective with cevian triangle of incenter.
    Perspector: X(55)
  5. A'B'C' perspective with Feuerbach triangle.
    Perspector: U=((b+c-a) (a^3(b+c)^2 -a^2(b-c)^2(b+c)- a(b-c)^2(b^2+6b c+c^2)+ (b-c)^4(b+c)): ... : ...)
    with search number: (2.5218174588907384, -10.902294391703143, 10.024491002829291)
  6. A'B'C' perspective with Extangents Triangle.
    Perspector: X(55)
  7. A'B'C' perspective with Apollonius triangle.
    Perspector: W=( a^2(a-b-c) (a^4(b+c)^2-2a^2(b+c)^2(b^2+c^2)-4a b c(b^3+c^3)+(b^3-b^2c+b c^2-c^3)^2):...:...)
    with search number: (3.1433391928115959, 2.7611853200434348, 0.27830270981049537)