Conjugate and Perpendicular Lines

Barycentric coordinates are used in respect of a triangle ABC. Let P = (u : v : w) be a point not on a sideline of ABC and let d=(p:q:r) be a line other than the side lines of ABC. The P-isoconjugate of d is the line


Let PaPbPc be the cevian triangle of P; let D, E, F be the points where d meets the sidelines BC, CA, AB; let D', E', F' be the points where d' meets the sidelines BC, CA, AB, then we obtain the following relations between the cross-ratios:

(B C P_a D)=(C B P_a D'),    (C A P_b E)=(A C P_b E'),    (A B P_c F)=(B A P_c F').

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If the lines d and d' are P- isoconjugate and perpendicular, then the point of intersection of d and d' is on bicevian conic C(P,H) , where H is the orthocenter of ABC.

Let Q be a point not on a sideline of ABC. Let d be a line through Q and d' its P- isoconjugate, then as d varies the locus of the point of intersection of d and d' is a cubic L(P,Q) through Q (which are double) and vertices of cevian triangle of P and Q . There are three pairs of P- isoconjugate and perpendicular lines (d,d') which intersec in three points of intersection (other than Pa,Pb and Pc ) of C(P,H) and L(P,Q).

Consider now the point Q on the bicevian conic C(P,H), we denote the point of intersection (other than Pa, Pb, Pc and Q ) of the C(P,H) and L(P,Q) by S[P,Q]. We call this the "sixth intersection" of the bicevian conic C(P,H) with the cubic L(P,Q).
The line d_1 passing through Q and S[P,Q] is perpendicular to its P- isoconjugate line d'_1 , and the point of intersection of d_1 and d'_1 is S[P,Q]. Let Q' be the second point of intersection of d'_1 with C(P,H) then S[P,Q]=S[P,Q'].
The tangents d_2 and d_3 in Q to L(P,Q) are perpendicular and P- isonconjugate to each other.

Graphic EPS