HYACINTHOS 29068

[Antreas P. Hatzipolakis]:

 

 

Let ABC be a triangle.

Denote:

Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.
Oa, Ob, Oc = the circumcenters of IBC, ICA, IAB, resp.
O1, O2, O3 = the reflections of Oa, Ob, Oc in BC, CA, AB, resp.
Ma, Mb, Mc = the midpoints of NaO1, NbO2, NcO3, resp.

ABC, MaMbMc are circumcyclologic.
ie the circumcircles of AMbMc, BMcMa, CMaMb and ABC are concurrent
the circmcircles of MaBC, MbCA, McAB and MaMbMc are concurrent

Cyclologic centers?

*****************

O1, O2, O3 are the reflections of I in Na, Nb, Nc, resp.

GENERALIZATION:

Let ABC be a triangle.

Denote: 

Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp,

A;', B', C' = points on INa, INb, INc, resp.  such that:

A'Na / A'I = B'Nb./ B'I = C'Nc / C'I = t

ABC, A'B'C' are circumcyclologic.

(for t = 1/3 the above)

Which are the cyclologic centers in terms of t ?

APH
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[Ercole Suppa]:

The cyclologic centers in terms of t are:

Z1 = Z(ABC,A'B'C')= -a (a-b) (a-c) (a^3 (1+2 t)+(b-c)^2 (b+c) (1+2 t)+a^2 (-c (1+2 t)+b (3+6 t+4 t^2))+a (-c^2 (1+2 t)-b c (1+2 t+4 t^2)+b^2 (3+6 t+4 t^2))) (a^3 (1+2 t)+(b-c)^2 (b+c) (1+2 t)+a^2 (-b (1+2 t)+c (3+6 t+4 t^2))-a (b^2 (1+2 t)+b c (1+2 t+4 t^2)-c^2 (3+6 t+4 t^2)))  : : (barys)

Z2 = Z(A'B'C', ABC) = (b+c-2 a t+2 b t+2 c t) (-a^3 (1+2 t)+(b^2-c^2) (b+c+2 c t)+a (1+2 t) (b^2+b c (-1+2 t)-c^2 (1+2 t))-a^2 (b+c (1+2 t)^2)) (a^3 (1+2 t)+(b^2-c^2) (b+c+2 b t)+a^2 (c+b (1+2 t)^2)+a (1+2 t) (-c^2+b^2 (1+2 t)+b (c-2 c t)))  : : (barys)


*** Particular case: for t = 1/2 we have


Z1 = X(1)X(28535)∩X(101)X(14422) 

= a (a-b) (a-c) (2 a^2+5 a b+2 b^2-4 a c-4 b c+2 c^2) (2 a^2-4 a b+2 b^2+5 a c-4 b c+2 c^2) : : (barys)

= lies on the circumcircle and these lines: {1,28535}, {101,14422}, {103,10246}, {244,28317}, {2291,4860}, {3576,28159}, {5425,28471}, {14413,28899}

=(6-9-13)  search numbers:  [0.27805490094376077337,-0.35949955332708227431,3.76121575685216185916]


Z2 = X(2)X(1155)∩X(4671)X(27756)

= (a-2 b-2 c) (2 a^2+2 a b+2 b^2-a c-b c-c^2) (2 a^2-a b-b^2+2 a c-b c+2 c^2) : : (barys)

= lies on these lines: {2,1155}, {4671,27756}, {4870,27761}, {4945,27741}, {4956,27753}, {5219,27737}, {5235,27740}, {5603,28537}, {9093,28899}

= isogonal conjugate of X(19654)

= barycentric quotient of X(i) and X(j) for these {i,j}: {6,19654}, {45,5220}, {3679,17294}, {4777,28898}, {28899,4588}

= trilinear product of X(i) and X(j) for these {i,j}: {4791,28899}

= trilinear quotient of X(i) and X(j) for these {i,j}: {1,19654}, {3679,5220}, {4671,17294}


=(6-9-13)  search numbers:  [-0.86360384156722765501,-0.79199874038912027765,4.58755769059249052073]


Best regards,
Ercole Suppa