(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:

A'B'C' perspective with ABC.
Perspector: X(354)

A'B'C' perspective with excentral triangle.
Perspector: X(57)

A'B'C' perspective with Cevian triangle of Gergonne point (the perspector is the same point with 2.)
Perspector: X(57)

A'B'C' perspective with cevian triangle of incenter.
Perspector: X(55)

A'B'C' perspective with Feuerbach triangle.
Perspector:
U=((b+ca) (a^3(b+c)^2 a^2(bc)^2(b+c) a(bc)^2(b^2+6b c+c^2)+ (bc)^4(b+c)): ... : ...)
with search number:
(2.5218174588907384, 10.902294391703143, 10.024491002829291)

A'B'C' perspective with Extangents Triangle.
Perspector: X(55)

A'B'C' perspective with Apollonius triangle.
Perspector:
W=( a^2(abc) (a^4(b+c)^22a^2(b+c)^2(b^2+c^2)4a b c(b^3+c^3)+(b^3b^2c+b c^2c^3)^2):...:...)
with search number:
(3.1433391928115959, 2.7611853200434348, 0.27830270981049537)