Δευτέρα, 5 Δεκεμβρίου 2011

Reflections of the Circumcircle in the sidelines


Let ABC be a triangle, (O1),(O2), (O3) the reflections of (O) in BC,CA,AB, resp., (O12),(O13) the reflections of (O1) in AC, AB, resp., (O21),(O23) the reflections of (O2) in BC, BA, resp. and (O31),(O32) the reflections of (O3) in CB, CA, resp.


The radical axes of [(O12),(O13)],[(O21),(O23)], [(O31),(O32)] concur at the isogonal conjugate of the Nine Point Circle Center X(54).

APH, 5 December 2011

Generalization:

Let ABC be a triangle, P a point, P1,P2,P3 the reflections of P in BC,CA,AB, resp., P12,P13 the reflections of P1 in in AC, AB, resp., P21,P23 the reflections of P2 in BC, BA, resp. and P31,P32 the reflections of P3 in CB, CA, resp.

Which is the locus of P such that the perpendicular bisectors of P12P13,P21P2, P31P32 are concurrent?

APH, 5 December 2011



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