Cyclology with medial triangle (Anopolis #2149 by Seiichi Kirikami)

Anopolis message #2149, by Seiichi Kirikami, Dec 11/2014 (https://groups.yahoo.com/neo/groups/Anopolis/conversations/topics/2149)

Let ΔA’B’C’ be the medial triangle of ΔABC.

Let X be the X(n)-ETC-center-of- ΔABC. Let A” be the X(n)-ETC-center-of-ΔAB’C’, B” be the X(n)-ETC-center-of-ΔBC’A’ and C” be the X(n)-ETC-center-of-ΔCA’B’.

The triangles ΔA’B’C’ and ΔA”B”C” are cyclologic (*)

(*) i.e., {a’} = circumcircle(ΔA’B”C”), {b’} = circumcircle(ΔB’C”A”) and {c’} = circumcircle(ΔC’A”B”) concur at Z’ 
and 
{a”} = circumcircle(ΔA”B’C’), {b”} = circumcircle(ΔB”C’A’) and {c”} = circumcircle(ΔC”A’B’) concur at Z”.

If X= u : v :w (trilinears) then:

1)     Z’ = u*(v*cos(B)-w*cos(C))*(u*(w*sin(B)-v*sin(C))+v*w*sin(B-C)) : :

2)     Z” = f(a,b,c,u,v,w)*g(a,b,c,u,v,w) : :

where:

f(a,b,c,u,v,w)= (a*b*c*v*w+c*(a^2-c^2)*w*u+b*(a^2-b^2)*u*v+a*(-b^2-c^2+a^2)*u^2)

g(a,b,c,u,v,w)= (-b*c *(v^2+w^2)*u +a*(b*v+c*w) *v*w +(2*a^2-b^2-c^2)*u*v*w)

3)     Z’ lies on circumcircle of ΔA’B’C’  and Z”  lies on circumcircle of ΔA”B”C” 

4)     Z’ and Z” are symmetrical w/r to the radical trace of circumcircles of ΔA’B’C’ and ΔA”B”C”

5)     Z”=midpoint(X, anticomplement(Z’)) or Z’=complement(reflection(X,Z”)).

Note:  anticomplement(Z’) lies on the circumcircle of  ΔABC.

6)     If {h(T)} is the rectangular circum-hyperbola with center T on the NPC of ΔABC, then for all P in {h(T)} we have Z’(P) = T, i.e.:

Z’(P)=X(11)  for all P on Feuerbach hyperbola

Z’(P)=X(115) for all P on Kiepert hyperbola

Result (6) above let us deduce the existence of some rectangular circum-hyperbolas:

Center	Trilinear equation [Through ETC-centers]	Perspector's 1st coordinate Properties ETC 6-9-13 search
X(116)	Σ a*(b-c)*(-b^2+a*b-b*c+c*a-c^2)*v*w [4, 103, 947, 1002, 1126, 1174, 2141, 3681, 3730, 4184]	a*(b-c)*(-b^2+a*b-b*c+c*a-c^2) Complement of X(3261) -6.381084332890005
X(120)	Σ (c^3+c*a^2-c*b^2-2*a*b*c+b*a^2+b^3-b*c^2)*v*w [4, 668, 1292, 4236]	X(3290)
X(122)	Σ b*c*(b^2-c^2)*(3*a^4-(b^2-c^2)^2-2*a^2*(b^2+c^2))*v*w [4, 20, 253, 1249, 1294, 3346, 3668, 5930]	(3*a^4-(b^2-c^2)^2-2*a^2*(b^2+c^2))*(b^2-c^2)/a Complement of X(3265)  -3.600719569463247
X(123)	Σ(c-b)*(a^4+2*b*c*a^2-2*a*b*c*(b+c)-(b^2-c^2)^2)*v*w [4, 961, 998, 1295, 1766, 2995, 3436]	(b-c)*(a^4+2*b*c*a^2-2*b*c*(b+c)*a-(b^2-c^2)^2) (6,2431)∩(230,231)  -3.012216446340097
X(124)	Σ (b-c)*(-b^3+b*a^2-a*b*c-c^3+c*a^2)*v*w*a [4, 58, 102, 573, 959, 994, 3417, 3869, 4225]	a*(b-c)*(-b^3-a*b*c-c^3+(b+c)*a^2) (6,652)∩(230,231) -3.443760727600679
X(5514)	Σ (c-b)*(a^3+a^2*(b+c)-(b-c)^2*(b+c)-a*(b+c)^2)*v*w [4, 40, 57, 189, 196, 223, 329, 937, 972, 1817, 2184, 3194, 3345]	X(6129)
X(5515)	Σ b*c*(b-c)*(a^2+(b+c)^2)*v*w [4, 75, 388, 1010, 1065, 1220, 2345, 4385]	(b-c)*(a^2+(b+c)^2)/a (230,231)∩(513,3700)
X(5521)	Σ (b-2)*(a^2-b^2+c^2)*(a^2+b^2-c^2)*v*w [4, 19, 28, 34, 286, 915, 1118, 1119, 5317]	(b-c)/(b^2+c^2-a^2) Isogonal conjugate of X(1332) -0.772000939862511

Some triads of (N, I, J) such that Z’=X(I) and Z”=X(J) for n=N:

(1, 11, 214), (2, 115, 2482), (3, 125, 1511), (5, 137, Q5), (6, 125, Q6), (8, 11, 1145), (9, 11, Q9)

where:

Q5= (1+2*cos(2*B)+2*cos(2*C)-2*cos(2*A))*

        (2*cos(A)*(cos(2*B-2*C)-2+4*cos(A)^2)+(-  1+4*cos(A)^2)*cos(B-C)) : :

     = On lines (2,1263), (3,2888), (5,930), (30,128), (137,3628) (at least)

     = On loci K038, K067

     = Complement of X(1263)

     = Inverse of X(5898) in circumcircle

     = Midpoint of: (5,930) (at least)

     = Reflection of (137/3628) (at least)

     = [7.346237636455233551025, 4.968703721274385706387, -3.189778541954149563316]

 Q6= a*(a^4-b^4-c^4+b^2*c^2)*(2*a^2-b^2-c^2)::

     = On lines (2,67), (3,1177), (5,542), (6,110), (74,5085), (113,1503), (125,3589) (at least)

     = On loci: K042, K043, K565

     = Complementary conjugate of X(858)

     = Complement of X(67)

     = Midpoint of: (6,110) (at least)                               

     = Reflection of: (125/3589), (141/5972) (at least)

     = [0.9502357345836802528899, 0.2943117751284885726792, 2.998339837010649974545]

Q9 = (2*a^2-a*(b+c)-(b-c)^2)*(a^2-2*a*(b+c)+b^2+b*c+c^2)::

      = On lines (1,3939), (2,3254), (3,2801), (9,100), (10,528), (119,516), (142,3035) (at least)

      = Complement of X(3254)

      = Midpoint of: (9,100) (at least)

      = Reflection of: (142/3035) (at least)

      = [3.086904417102852112698, 0.3728597156591479833209, 1.957805717019037216497]