HYACINTHOS 10888

Dear Antreas and everyone,

[APH]

>I am wondering what geometrical properties [locus, envelope] has
>the "dual" of the MacBeath inconic ie the MacBeath circumconic
>[= the conic centered at N, and passing through A,B,C.]

My limited understanding of a dual comes from here
http://www.math.fau.edu/yiu/GeometryNotes020402.ps page 120.
By my reckoning, I think the dual is the circumconic,
a^2 SA y z + cyclic = 0, centered on X(6) passing through
110, 287, 648, 651, 677, 895, 1331, 1332, 1797, 1813, 1814, 1815

If it is of interest, here are a few centers, other than X(399,
1312, 1313), that I think are on the MacBeath conic,
a^4 SA^2 x^2 - 2 b^2 c^2 SB SC y z + Cyclic = 0.

rfl X(339) in X(5)
a^2 SB SC (SB SC - SA^2)^2
on lines {3,112},{4,147},{25,110},{114,132}

rfl X(339) in Euler line.
a^2 SA (b^2 - c^2)^2 (S^2 + SA^2 - 4 SB SC)^2

(b - c)^2 (b + c - a)^2 SA
on lines {3, 8}, {11, 123}, {116, 122}

(b - c)^2 SB SC
on lines {4, 145}, {25,105},{124,136}

b^2 c^2 (b^2 - c^2)^2 SB SC
on lines {4,94},{25,98},{115,135},{125,136}

a^2 (b^2 - c^2) SB SC (SB - SC)
on lines {4,147},{25,111},{127,136}

b^2 c^2 SA (SB - SC)^2
on lines {3,76},{115,127}

a^2 SA^3 (SB - SC)^2
on lines {3,74},{0,122,125},{0,127,136}

b^2 c^2 (b - c)^2 SB SC
on line {4, 150}

a^2 SA (a^2 (SB SC - SA^2) + SA (SB - SC)^2)^2
on lines {3,74},{113,131}

b^2 c^2 SB SC (S^2 - 3 SB SC)^2
on lines {3,107},{113,133}

SA (a b c (b + c) - 2 S^2 + a (b + c) (b c - 2 SA) - 2 SB SC)^2
on lines {3,100},{117,131}

note that many of these are intersections of {Focus, NP},{NP,NP}

In general, a point on the conic has 3 friends, its reflection in X
(5) and their reflection in the Euler line.

Also, I find here
http://www.genealogy.ams.org/html/id.phtml?id=24339
an entry for Alexander Murray MacBeath. The same one?


Best regards,
Peter Moses