HYACINTHOS 26358

[Le Viet An] (*):

Let ABC be a triangle and L a line passing through O.

Denote:

A',B',C' = the orthogonal projections of A,B,C on L, resp. 

 

La, Lb, Lc = the parallels though A', B', C' to BC,CA,AB, resp.

A*B*C* = the triangle bounded by La, Lb, Lc

 

The NPC of A*B*C* touches the circumcircle of ABC.

For L = Euler line, the touchpoint is X110.

For L = X74OX110 which is the touchpoint?

For L = OK (Brocard axis) which is it?

(*) 9-circle touches circumcircle  

[Peter Moses]:

Hi Antreas,

For an line L = OP{p,q,r}, the touchpoint, T, is
T = a^2(b^2 (p - r) + (a^2 - c^2) q) (c^2 (p - q) + (a^2 - b^2) r):: = the isogonal conjugate of the orthopoint of (L / \ infinity).

For P = X(74), T = X(476),
P = K (Brocard), T = X(99).
P = I, T = X(100).

 

Best regards,
Peter Moses.