ORTHOLOGIC TRIANGLES

Let ABC be a triangle, P, Q two points, A'B'C' the padal triangle of Q and S a point on the Euler line such that OS/OH= t: number.

Denote:

Ab, Ac = the orthogonal projections of A on BP, CP, resp.
Sa = same to S point of the triangle A'AbAc.
Similary Sb,Sc.

Cases the triangle ABC, SaSbSc are orthologic.

1. Let ABC be a triangle, P a point and A'B'C' the pedal triangle of a point Q.

Denote:

Ab, Ac = the orthogonal projections of A on BP, CP, resp.

Ga = the centroid of A'AbAc.
Similarly Gb, Gc

ABC, GaGbGc are circumorthologic.
ie The Orthologic center (ABC, GaGbGc) = X1 lies on the circumcircle of ABC.
The Orthologic center (GaGbGc, ABC) = X2 lies on the circumcircle of GaGbGc.

2. Let ABC be a triangle, P = I = X(1), A'B'C' the pedal triangle of a point Q and S a point on the Euler line such that OS/OH= t: number.

Denote:

Ab, Ac = the orthogonal projections of A on BI, CI, resp.

Sa = same to S point of the triangle A'AbAc.
Similarly Sb, Sc

ABC, SaSbSc are Orthologic.

3. Let ABC be a triangle, P, Q two isogonal conjugate points, A'B'C' the pedal triangle of Q and S a point on the Euler line such that OS/OH= t: number.

Denote:

Ab, Ac = the orthogonal projections of A on BP, CP, resp.

Sa = same to S point of the triangle A'AbAc.
Similarly Sb, Sc

ABC, SaSbSc are circumoerthologic. ie The Orthologic center (ABC, SaSbSc) = X1 lies on the circumcircle of ABC.
The Orthologic center (SaSbSc, ABC) = X2 lies on the circumcircle of SaSbSc.