ANOPOLIS 326

Antreas P. Hatzipolakis

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

Denote:

L11 = the perpendicular line to AA' at A'

L22 = the perpendicular line to BB' at B'

L33 = the perpendicular line to CC' at C'


L12 = the reflection of L11 in BB'

L13 = the reflection of L11 in CC'


M12 = the parallel to L12 through B'

M13 = the parallel to L13 through C'

A" = line M12 /\ line M13

Similarly B" and C".

O1 = the circumcenter of the triangle A"B'C'

O2 = the circumcenter of the triangle B"C'A'

O3 = the circumcenter of the triangle C"A'B'

Conjecture 1:

The points I, O1, O2, O3 are concyclic. Center of the circle?

Conjecture 2: The circumcircles (O1), (O2), (O3) are concurrent.
Point of concurrence?

Figure:
http://anthrakitis.blogspot.gr/2013/05/concurrent-circles.html

APH