HYACINTHOS 25609

[Antreas P. Hatzipolaks]:

 

Let ABC be a triangle.

Denote:

A', B', C' = the reflections of O in BC, CA, AB, resp.

[A', B', C' are the reflections of A,B,C in N, resp.]

 

Ab, Ac = the orthogonal projections of A' on AC, AB, resp.

 

(Na) = the NPC of AAbAc. Similarly Nb, Nc.

(Na), (Nb), (Nc) are concurrent.

Point of concurrence ?

GENERALIZATION:

Let ABC be a triangle and P a point.

Denote:

A', B', C' = the reflections of A,B,C in P, resp.

Ab, Ac = the orthogonal projections of A' on AC, AB, resp.

Bc, Ba = the orthogonal projections of B' on BA, BC, resp.

Ca, Cb = the orthogonal projections of C' on CB, CA, resp.

 

(Na), (Nb), (Nc) = the NPCs of AAbAc, BBcBa, CCaCb, resp.

(Na), (Nb), (Nc) are concurrent.

 

Point of concurrence in terms of P?

[Peter Moses]:

Hi Antreas,

 

If P = {p,q,r}, then the point of concurrence is

Q = a^2 (2 a^2 b^2 c^2 p+a^4 c^2 q+b^4 c^2 q-2 a^2 c^4 q-2 b^2 c^4 q+c^6 q+a^4 b^2 r-2 a^2 b^4 r+b^6 r-2 b^4 c^2 r+b^2 c^4 r) (b^2 c^2 p^2+a^2 c^2 p q-c^4 p q+a^2 b^2 p r-b^4 p r+a^4 q r-a^2 b^2 q r-a^2 c^2 q r)::

 

Some ETC examples: {P,Q}

{{1,11570},{4,1986},{5,11557}, {6,5477},{15,6783},{16,6782},{ 23,3580},{36,1737},{110,7471}, {186,403},{187,230},...}.

 

X(2) :  a^2 (a^4-b^4+b^2 c^2-c^4) (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2+2 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6)::
on lines {{2,2781},{23,6593},{25,110},{ 51,542},{52,10294},{74,9818},{ 113,403},{125,5133},...}}.

 

X(20) : a^2 (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2+2 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6) (a^8-2 a^6 b^2+2 a^2 b^6-b^8-2 a^6 c^2+7 a^4 b^2 c^2-4 a^2 b^4 c^2-b^6 c^2-4 a^2 b^2 c^4+4 b^4 c^4+2 a^2 c^6-b^2 c^6-c^8)::
on lines {{2,974},{3,74},{22,9934},{69, 146},{113,403},{125,5907},...} }.

 

X(21): a (a+b) (a+c) (a^4-b^4+a^2 b c-a b^2 c-a b c^2+2 b^2 c^2-c^4) (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2+2 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6)::
on lines {{21,2778},{28,110},{113,403}, ...}}.

 

X(22): (a^4 b^2-b^6+a^4 c^2-2 a^2 b^2 c^2+b^4 c^2+b^2 c^4-c^6) (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2+2 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6)::
on lines {{2,98},{5,12099},{113,403},.. .}}.

 

X(25): (2 a^2-b^2-c^2) (a^2+b^2-c^2) (a^2-b^2+c^2) (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2+2 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6)::
on lines {{4,541},{25,542},{51,125},{10 7,11005},{110,6353},{112,6792} ,{113,403},...}}.

 

X(32): (a^2-b c) (a^2+b c) (2 a^4-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4)::
on lines {{6,98},{32,2782},{39,12042},{ 99,3053},{114,230},{115,546},. ..}}.

 

X(39): (a^4+a^2 b^2-b^4+a^2 c^2-b^2 c^2-c^4) (2 a^4-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4)::
on lines {{6,147},{30,1569},{98,3815},{ 99,7762},{114,230},{115,3850}, ...}}.

 

X(55): (2 a^2-a b-b^2-a c+2 b c-c^2) (a^3 b-a^2 b^2-a b^3+b^4+a^3 c+a b^2 c-a^2 c^2+a b c^2-2 b^2 c^2-a c^3+c^4)::
on lines {{11,118},{12,5884},{57,5660}, {63,3035},{80,11529},{100,3474 },{119,912},...}}.

 

X(56): (2 a-b-c) (a+b-c) (a-b+c) (a^3 b-a^2 b^2-a b^3+b^4+a^3 c+a b^2 c-a^2 c^2+a b c^2-2 b^2 c^2-a c^3+c^4)::
on lines {{1,6713},{10,5083},{11,65},{1 2,5883},{46,5840},{56,952},{57 ,80},{78,3035},{100,1788},{104 ,1470},{109,6788},{119,912},.. .}}.

 

X(99): a^2 (a-b) (a+b) (a-c) (a+c) (a^4 b^4-2 a^2 b^6+b^8-2 b^6 c^2+a^4 c^4+4 b^4 c^4-2 a^2 c^6-2 b^2 c^6+c^8)::
on lines {{4,69},{99,512},{112,249},... }}.

 

Best regards,

Peter Moses.