Dear Antreas, Fred and Darij [FL] ... Consider three lines La, Lb, Lc going respectively through A, B, C and such as (La,POa) = (Lb,POb) = (Lc,POc) = phi where...
Dear Juan Carlos and fellow Hyacinthians, ... triangle. ... Z ... The proof immediately follows from the concurrence of altitudes in a triangle and certain...
Dear Jean-Pierre, Antreas and Darij ... That's fine. A link between nK0 and pK. I have the pencil MacCay -4s tan(phi) Kjp. This cubic is an isocubic iff phi =...
Dear Fred, Darij and Antreas ... C ... is ... = ... and ... This is the same cubic with -phi instead of phi - ie the isogonal conjugate of the locus of P -. Of...
Dear Jean-Pierre and Fred, ... All the cubics of the pencil are K60+ with three asymptotes concurring at G. There is only one K60++ in the pencil, the one...
Dear friends, let Q be a fixed point and phi the measure (mod. pi) of an angle. the locus of P such as the sum of line angles (AQ,AP)+(BQ,BP)+(CQ,CP) = phi is...
Let ABC be a triangle and P a point. Let AaBaCa be the orthic triangle of PBC [Aa on BC, Ba on CP, Ca on PB] and BbCbAb, CcAcBc the orthic triangles of PCA,...
Dear Jean-Pierre and Bernard, ... I arrive at the same expression for phi: phi = -(a^2-b^2)(b^2-c^2)(c^2-a^2)/(a^2b^2+b^2c^2+c^2a^2)/s Good evening. Fred...
Dear Antreas ... The circumcircle + the Mac Cay cubic ... The circumcircle ... The Darboux cubic ... The Kjp cubic + the circumcircle ... The whole plane ...
... This is trivial, and trivial is also the case of the circumcenters. But how about other points (I, G, K, etc)? ie if Qa, Qb, Qc are the Q-points of PBC,...
... [## 1 - 4 answered by ND, JPE] Denote Pa, Pb, Pc = the orthogonal projections of P on the lines BaCa, CbAb, AcBc, resp. Which is the locus of P such that ...
Perhaps this is obvious to many of you; Can you please suggest how to construct a cyclic quadrilateral given the lengths of its four sides? Is there a solution...
Thanks Alexey. However,as my message indicated, I want to know whether a solution exists which does not calculate arithmetically the radius. Is there any such...
... given ... the ... This problem is owned by the mathematician Sturm who has borned in Genova in 1803. Suppose the problem has been solved, draw a cyclic...
The following paper has been published in Forum Geometricorum. It can be viewed at http://forumgeom.fau.edu/FG2004volume4/FG200401index.html The editors Forum...
Let ABC be a triangle P a point and PaPbPc the pedal triangle of P. Let Ha, Hb, Hc be the orthocenters of the triangles APbPc, BPcPa, CPaPb, resp. Which is the...
Let ABC be a triangle P a point and PaPbPc the pedal triangle of P. Let Oa, Ob, Oc be the circumcenters of the triangles APbPc, BPcPa, CPaPb, resp. Which is...
Dear Antreas, [APH]: Let ABC be a triangle P a point and PaPbPc the pedal triangle of P. Let Ha, Hb, Hc be the orthocenters of the triangles APbPc, BPcPa,...
Dear Antreas, [APH]: Let ABC be a triangle P a point and PaPbPc the pedal triangle of P. Let Oa, Ob, Oc be the circumcenters of the triangles APbPc, BPcPa,...
Let ABC be a triangle and (Ma), (Mb), (Mc) three circles such that none lies inside another. Define as Monge line of the circles the line passing through the...
Dear Antreas, [APH]: Let ABC be a triangle and (Ma), (Mb), (Mc) three circles such that none lies inside another. Define as Monge line of the circles the line...
Dear Antreas, [APH]: Let ABC be a triangle P a point and PaPbPc the pedal triangle of P. Let Oa, Ob, Oc be the circumcenters of the triangles APbPc, BPcPa,...
Let ABC be a triangle P a point and PaPbPc the pedal triangle of P. Let Ea, Eb, Ec be the Euler lines of the triangles APbPc, BPcPa, CPaPb, resp. Which is the...
Let ABC be a triangle P,P* two isogonal conjugate points and PaPbPc, P*aP*bP*c their pedal triangles. Let Oa, Ob, Oc be the circumcenters of the triangles ...
Dear Antreas and Paul, [APH]: Let ABC be a triangle P a point and PaPbPc the pedal triangle of P. Let Ha, Hb, Hc be the orthocenters of the triangles APbPc,...
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