## Δευτέρα, 25 Μαρτίου 2013

### RADICAL CENTERS, CIRCUMCIRCLE, EXCIRCLES

Let ABC be a triangle and Ia,Ib,Ic the excenters and O the circumcenter.

1.

Denote:

r1 = the rdical center of (O),(Ib),(Ic)

r2 = the rdical center of (O),(Ic),(Ia)

r3 = the rdical center of (O),(Ia),(Ib)

Perspective Triangles (?):

1.1. ABC, r1r2r3

1.2. IaIbIc, r1r2r3

2.

Denote:

Ja = the excenter of the excircle respective to BC of the triangle OBC

Jb = the excenter of the excircle respective to CA of the triangle OCA

Jc = the excenter of the excircle respective to AB of the triangle OAB

R1 = the rdical center of (O),(Jb),(Jc)

R2 = the rdical center of (O),(Jc),(Ja)

R3 = the rdical center of (O),(Ja),(Jb)

Perspective Triangles (?):

2.1. ABC, R1R2R3

2.2. JaJbJc, R1R2R3

3.

Denote:

i1 = the radical center of (Ja),(Ib),(Ic)

i2 = the radical center of (Jb),(Ic),(Ia)

i3 = the radical center of (Jc),(Ia),(Ib)

j1 = the radical center of (Ia),(Jb),(Jc)

j2 = the radical center of (Ib),(Jc),(Ja)

j3 = the radical center of (Ic),(Ja),(Jb)

Perspective triangles (?):

3.1. ABC, i1i2i3

3.2. ABC, j1j2j3

3.3. i1i2i3, j1j2j3

3.4. i1i2i3, IaIbIc

3.5. i1i2i3, JaJbJc

3.6. j1j2j3, IaIbIc

3.7. j1j2j3, JaJbJc

3.8. IaIbIc, JaJbJc

Antreas P. Hatzipolakis, 25 March 2013