ANOPOLIS 320

Antreas P. Hatzipolakis

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

Denote:

1 = the perpendicular line to AA' at A'
2 = the perpendicular line to BB' at B'
3 = the perpendicular line to CC' at C'
(ie the lines 1,2,3 bound the antipedal triangle of I wrt A'B'C')

Now let's reflect 1,2,3 in AA', BB', CC':

11 = the reflection of 1 in AA' (it is identical to 1)
12 = the reflection of 1 in BB'
13 = the reflection of 1 in CC'

21 = the reflection of 2 in AA'
22 = the reflection of 2 in BB' (it is identical to 2)
23 = the reflection of 2 in CC'

31 = the reflection of 3 in AA'
32 = the reflection of 3 in BB'
33 = the reflection of 3 in CC' (it is identical to 3)

O1 = the circumcenter of the triangle bounded by
the lines (11,12,13)
O2 = the circumcenter of the triangle bounded by
the lines (21,22,23)
O3 = the circumcenter of the triangle bounded by
the lines (31,32,33)

Conjecture:
O1, O2, O3 and O [circumcenter of ABC] are concyclic.
(Otherwise the Center of the circle (O1O2O3) might be interesting
if it is lying on some interesting lines or curves.)

Locus of variable P (instead of I) with that property?

APH