[APH]:

>Recent addition in Bernard's site:

>

>6. Let P be a point with traces A_P, B_P, C_P. Denote by L1 the line

>joining the perpendicular feet of A_P on the cevians BB_P and CC_P.

>Let L2, L3 be the two lines analogously defined. The triangle bounded

>by L1, L2, L3 is perspective with ABC if and only if P lies on the

>Darboux cubic (together with the line at infinity, the three circles

>with diameters BC, CA, AB, and the Yiu quintic Q006 ((Paul Yiu,

>Hyacinthos #1967).

>

> http://perso.wanadoo.fr/bernard.gibert/Exemples/k004.html

>

>Now, denote

>

>A* = L1 /\ BC

>

>B* = L2 /\ CA

>

>C* = L3 /\ AB

>

>Which is the locus of P such that A*,B*,C* are collinear?

Variations:

1. Let A'B'C' be the cevian triangle of P.

Denote

Ab = CC' /\ Parallel to BB' from A'

Ac = BB' /\ Parallel to CC' from A'

L1 = The line AbAc. Similarly the lines L2, L3.

Which is the locus of P such that the triangle bounded by L1,L2,L3

is perspective with ABC?

[Special case: L1,L2,L3 are concurrent]

2. Let A'B'C' be the cevian triangle of P.

Denote

Ab = CC' /\ Parallel to AB from A'

Ac = BB' /\ Parallel to AC from A'

L1 = The line AbAc. Similarly the lines L2, L3.

Which is the locus of P such that the triangle bounded by L1,L2,L3

is perspective with ABC?

[Special case: L1,L2,L3 are concurrent]

Also same questions for A*,B*,C* as defined above.

Greetings from Athens

Antreas

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