Δευτέρα, 20 Μαΐου 2013

EXCENTERS - CIRCUMCENTERS - CONICS

Let ABC be a triangle and P a point.

Denote:

Iab, Iac = the excenters of IBC respective to angles PBC, PCB.

Ibc, Iba = the excenters of ICA respective to angles PCA, PAC.

Ica, Icb = the excenters of IAB respective to angles PAB, PBA.

For P = O, the six excenters are concyclic, lying on the circumcircle (O).

Are they always lying on a conic? And for which points P the conic is circle?

Denote:

Oa, O'a = the circumcenters of PIbcIcb, PIbaIca, resp.

Ob, O'b = the circumcenters of PIcaIac, PIcbIab, resp.

Oc, O'c = the circumcenters of PIabIba, PIacIbc, resp.

The triangles OaObOc, O'aO'bO'c are perspective.

The line segments OaO'a, ObO'b, OcO'c are bisected by the perspector P' of the triangles, therefore the six points lie on a conic with center P'.

For which points P the conic is circle?

Antreas P. Hatzipolakis, 20 May 2013

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