(Advanced Plane Geometry Dao Thanh Oai)

  Let ABC be a triangle, (O_e) be the circle touching to the circumcircle at A and touching the incircle at A_e. Define B_e, C_e cyclically. Let A'B'C' be the tangent line of the incircle at A_e, B_e, C_e. 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)