Σάββατο, 16 Μαρτίου 2013

REFLECTIONS OF PARALLELS TO OP LINE

1.

1. Let ABC be a triangle and A'B'C' the circumcevian triangle of I.

Let La, Lb. Lc be the parallels to OI line through A',B',C', resp. and Ma,Mb,Mc the reflections of La,Lb,Lc in AA', BB',CC', resp.

The lines Ma,Mb,Mc are concurrent.

Locus:

Let ABC be a triangle and A'B'C' the circumcevian triangle of point P.

Let La, Lb, Lc be the parallels to OP line through A',B',C', resp. and Ma,Mb,Mc the reflections of La,Lb,Lc in AA', BB',CC', resp.

Which is the locus of P such that the lines Ma,Mb,Mc are concurrent ?

2. Orthic Triangle version:

Let ABC be a triangle, A'B'C' the pedal triangle of H (orthic triangle) and A"B"C" the circumcevian triangle of H wrt A'B'C'.

Let La,Lb,Lc be the parallels to HN line (Euler line) through A",B",C", resp. and Ma,Mb,Mc the reflections of La,Lb,Lc in HA',HB',HC', resp.

The lines Ma,Mb,Mc are concurrent.

Locus:

Let ABC be a triangle, P a point, A'B'C' the pedal triangle of P and A"B"C" the circumcevian triangle of P wrt A'B'C'.

Let La,Lb,Lc be the parallels to PQ,, where Q is the center of the pedal circle of P, through A",B",C", resp. and Ma,Mb,Mc the reflections of La,Lb,Lc in the lines PA',PB',PC', resp.

Which is the locus of P such that Ma,Mb,Mc are concurrent?

Antreas P. Hatzipolakis, 16 March 2013

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