Write about X(1251) here.
As for X(1251), a / (Sqrt[3] S - 2 sb sc) :: and its friend, a mild saving grace .. they lie on the Feuerbach hyperbola. .. as I think does any point of the form a / (k(a,b,c) - 2 sb sc) ::
Best regards,
Peter.
Green = fissile, blue = strong, red = weak
X(1251), = g X(1082) = a / (b c - Sqrt(3) S - SA) ::
lives on lines ...
{{1,15,1276,1652,2306},
{4,1832},
{7,559},
{13 ,80}
{37,2153},
{55,199,1030,1631,1962,2160,2178,2294},
{1250,2153}}
Its partner a / (b c + Sqrt(3) S - SA) :: = g X(559)
lives on lines ...
{{1,16,1250,1277,1653},
{4,1833},
{7,554,1082},
{14,80},
{37,2154},
{55,199,1030,1631,1962,2160,2178,2294},
{90,2307}}
Best regards,
Peter.
X(199) = X(10)-CEVA CONJUGATE OF X(6)
X55 is proNagel
In the meantime in ETC I have found a real-world candidate (although,
unfortunately, many of the points in ETC have no relevance to
anything and are there only through the misfortune of their having
coordinates). It is the only point in Kimberling ETC with a root3 in
it (that I could find) that was on a weak line it is x1251. I
mentioned this point before and Peter (thanks Peter) sent me this
X(1251), = g X(1082) = a / (b c - Sqrt(3) S - SA) ::
lives on lines ...
...snip...
{55,199,1030,1631,1962,2160,2178,2294},
Its partner a / (b c + Sqrt(3) S - SA) :: = g X(559)
lives on lines ...
..snip...
{55,199,1030,1631,1962,2160,2178,2294},
When I get things like this I color strong points blue, fissile green, and weak red. The above lines are all red points !
199 is the Cevian quotient of So and K and 55 is the proNagel point. [I allow Cevian quotient, because it is projectively invariant and its name actually describes what it does).
I will deal with this point as a start because the algebra is easy
X1082 a b c - a Sqrt(3) S - a SA)
which mixes the incenter with abc - aSA a point on the GWo line, where Wo = aSA:: is one of the Clawson points.
So this point is a real world candidate. Now what is interesting about it?
ETC lists
X(1082) lies on these lines: 1,3 7,554 13,226 298,319
X(1082) = isogonal conjugate of X(1251)
So at least it has some properties, although I do not see how this point can be on IO. Oh, wait a second
bc - SA ~ -(b-c)^2 + aa = sbc so we can write this as
a sbc - a Z where Z = 2 r3 S, more or less the same Z as for the Fermats. This is on the incenter-coMittenpunkt line. Wow, too good to be true.
Note: a sbc = isogonal of the Mittenpunkt which I call the coMittenpunkt. It is the desmic mate of the Mittenpunkt.
So, this point is almost exactly of the form you (Wison Strohers) gave above (but with a different Z). We have a toy to play with !
This is not on the line IO so either ETC information is incorrect or Peter's translation from angles to distances.
With p+- = a sbc - a Z we should consider q+- = a soa - a Z
There is a real subject here! Remember there are 8 versions of these points.
Using soa + sbc = bc and soa - sbc = SA we have
p+ + p- ~ a sbc, a coMittenpunkt pt. p+ - p- ~ a, the incenter
q+ + q- ~ a soa, the Mittenpunkt. q+ - q- ~ a, the incenter
q+ + p- ~ abc, the centroid. q+ - p- ~ a SA + 2 a Z, a new point
q+ - p+ ~ a SA, a Clawson pt q+ + p+ ~ abc + 2 a Z, a new point
Here we try extraversions
a / (b c - r3 S - SA) : b / (ca - r3 S - SB) : c / (ab - r3 S - SC)
+-- +-- +--
which is of form uvw
now apply a-extraversion move
a / (b c - r3 S - SA) : b / (+ca + r3 S + SB) : c / (ab + r3 S + SC)
+-- +++ +++
which is of the form u V W
now apply b extra
a / (b c + r3 S + SA) : b / (ca + r3 S + SB) : c / (ab - r3 S - SC)
+++ +++ +--
Which is of the form U V w
abc = identity and ab = c
This is quartile (C2 x C2), so that point is 8 fold = C2 x C2 x C2.
Let r be the root extraversion
a / (b c + r3 S - SA) : b / (ca + r3 S - SB) : c / (ab + r3 S - SC)
++- ++- ++-
now do ra
a / (b c + r3 S - SA) : b / (ca - r3 S + SB) : c / (ab - r3 S + SC)
++- +-+ +-+
now do rar
a / (b c - r3 S - SA) : b / (ca + r3 S + SB) : c / (ab + r3 S + SC)
+-- +++ +++
hence rar = a so that r and a commute