HYACINTHOS 13877

Dear All My Friends,
Given reference triangle ABC and point P with barycentrics (x : y : z).
Construct some points as following:
gP = isogonal conjugate of P
tP = isotomic conjugate of P
tgP = isotomic conjugate of gP (isotomic of isogonal conjugate of P)
gtP = isogonal conjugate of tP (isogonal of isotomic conjugate of P)
Construct some intersections of lines:
Gt = intersection of lines PtP and gPgtP
Tg = intersection of lines PgP and tPtgP
Denote the map P to Tg as Tmap. It means Tmap(P) = Tg
Denote the map P to Gt as Gmap. It means Gmap(P) = Gt
It is interesting that six points A, B, C, gP, tP, Tg, Gt are on one conic, say (Co).
Moreover, conic (Co) is isogonal conjugate of line PgtP and isotomic conjugate with line PtgP.
We can calculate the barycentrics of Tmap and Gmap. The results as following:

Tmap(P) = (b^2 - c^2)*(b^2*x^2 - a^2*y^2)*(c^2*x^2 - a^2*z^2) / x : :
Gmap(P) = a^2*(b^2 - c^2)*(x^2 - y^2)*(x^2 - z^2) / x : :

When P are some proper points so these two maps can generate some nice points. Following are some of them:
1. P = X(31) = (a^3 : b^3 : c^3)
Tmap = (a^2 + b^2)*(a^2 + c^2) / a : :
Search value: 2.81798455442606
Gmap = (a^6 - b^6)*(b^2 - c^2)*(a^6 - c^6) / a : :
Search value: 1.97372432580138
2. P = X(32) = (a^4 : b^4 : c^4)
Tmap = (a^4 + a^2*b^2 + b^4)*(a^4 + a^2*c^2 + c^4) / a^2 : :
Search value: 3.07079726519694
Gmap = (a^8 - b^8)*(b^2 - c^2)*(a^8 - c^8) / a^2 : :
Search value: 2.14803340062874
3. P = X(561) = (1/a^3 : 1/b^3 : 1/c^3)
Tmap = (a^2 + b^2)*(a^4 + b^4)*(a^2 + c^2)*(a^4 + c^4) / a^3 : :
Search value: 3.33657836754355
Gmap = Gmap(X(31))
4. P = X(560) = (a^5 : b^5 : c^5)
Tmap = Tmap(X(561))
Gmap = (a^10 - b^10)*(b^2 - c^2)*(a^10 - c^10) / a^3 : :
Search value: 2.31435145982684
5. P = X(1502) = (1/a^4 : 1/b^4 : 1/c^4)
Tmap = (a^10 - b^10)*(a^10 - c^10) / ((a^2 - b^2)*(a^2 - c^2)*a^4)
Search value: 3.58368143130959
Gmap = Gmap(X(32))
6. P = X(1501) = (a^6 : b^6 : c^6)
Tmap = Tmap(X(1502))
Gmap = (a^12 - b^12)*(b^2 - c^2)*(a^12 - c^12) / a^4
Search value: 2.46459939393964
7. P = X(1928) = (1/a^5 : 1/b^5 : 1/c^5)
Tmap = (a^12 - b^12)*(a^12 - c^12) / ((a^2 - b^2)*(a^2 - c^2)*a^5) : :
Search value: 3.79939674508783
Gmap = Gmap(X(560))
8. P = X(1917) = (a^7 : b^7 : c^7)
Tmap = Tmap(X(1928))
Gmap = (a^14 - b^14)*(b^2 - c^2)*(a^14 - c^14) / a^5 : :
Search value: 2.59759070019078

So we have total ten new triangle centers.
Please kindly check and give me advice and correct for me if any wrong of these points.
Thank you and best regards,
Bui Quang Tuan