VIJAY-PAASCHE-HUTSON TRIANGLES

Let pa be the parabola with focus A and directrix BC. Let Ba be the point, closer to C, at which pa cuts AC and let Ca be the point, closer to B, at which which pa cuts AB. Build Cb, Ab and Ac, Bc cyclically. Points Ab, Ac, Bc, Ba, Ca, Cb lie on an ellipse named the Paasche ellipse and have trilinear coordinates: (1)

Ab = 0 : 2*R/b : 1,  Bc = 1 : 0 : 2*R/c,  Ca = 2*R/a : 1 : 0

Ac = 0 : 1 : 2*R/c,  Ba = 2*R/a : 0 : 1,  Cb := 1 : 2*R/b : 0

Let A'b, B'c, C'a be the midpoints of BAb, CBc, ACa, respectively, and A'c, B'a, C'b be the midpoints of CAc, ABa, BCb, respectively, i.e.: (2)

A'b = 0 : 1/b : 1/(4*R+c),  B'c = 1/(a+4*R) : 0 : 1/c,  C'a = 1/a : 1/(4*R+b) : 0

A'c = 0 : 1/(4*R+b) : 1/c,  B'a = 1/a : 0 : 1/(4*R+c),  C'b = 1/(a+4*R) : 1/b : 0

Also, define A"b, A"c, B"c, B"a, C"a, C"a as follows: (3)

A"b = reflection of Ab in B = 0 : 1/b : -1/(2*R+2*c),  B"c = reflection of Bc in C = -1/(2*R+2*a) : 0 : 1/c,  C"a = reflection of Ca in A = 1/a : -1/(2*R+2*b) : 0

A"c = reflection of Ac in C = 0 : -1/(2*R+2*b) : 1/c,  B"a = reflection of Ba in A = 1/a : 0 : -1/(2*R+2*c),  C"b = reflection of Cb in B = -1/(2*R+2*a) : 1/b : 0

The following table shows 31 triangles obtained from the above defined points. These triangles are named Vijay-Paasche-Hutson triangles. In this table, An, Bn, Cn refer to the vertices of the nth-Vijay-Paasche-Hutson triangle.

# A-vertex A-vertex trilinear coordinates
1 AcCa ∩ AbBa (-b*c+4*R^2)/a :

2*R+c :

2*R+b
2 CaBc ∩ CbBa (8*R^3-a*S)/a :

-c*a+4*R^2 :

-b*a+4*R^2
3 CbAc ∩ AbBc (-b*c+4*R^2)/(4*R^2) :

(2*R+b)/b :

(2*R+c)/c
4 A'cC'a ∩ A'bB'a (2*a*b*c+(b+c)*S)/S :

(b*a+S)/b :

(c*a+S)/c
5 C'aB'c ∩ C'bB'a (2*(a^6+2*(b+c)*a^5-(2*b^2-b*c+2*c^2)*a^4-4*(b+c)*(b^2+c^2)*a^3+(b^4+c^4-2*(b^2+9*b*c+c^2)*b*c)*a^2+2*(b^2-c^2)^2*(b+c)*a+(b^2-c^2)^2*b*c)*b*c+((b+c)*a^5+2*b*c*a^4-2*(b+c)*(b^2+c^2)*a^3-4*(b^2+8*b*c+c^2)*b*c*a^2+(b+c)*(b^2-4*b*c-c^2)*(b^2+4*b*c-c^2)*a+2*(b^2-c^2)^2*b*c)*S)/(4*S*(2*b*c+S)*a) :

-2*a*b*c-(a+c)*S :

-2*a*b*c-(a+b)*S
6 C'bA'c ∩ A'bB'c (2*a*b*c+(b+c)*S)*S/(2*b*c+S) :

(c*a+S)*(2*b*a+S)/b :

(b*a+S)*(2*c*a+S)/c
7 C'bAc ∩ B'cAb (a^4-2*(b-c)^2*a^2+(b^2-c^2)^2)*S/(4*(2*b*c+S)*a^2) :

c*a+S :

b*a+S
8 'aCa ∩ C'aBa (-a^4+2*(b^2+6*b*c+c^2)*a^2+8*(b+c)*S*a-(b^2-c^2)^2)/4*S*a) :

(b*a+S)/b :

(c*a+S)/c
9 A'cB'c ∩ A'bC'b (2*a*b*c+(b+c)*S)/(2*b*c+S) :

-(b*a+S)/b :

-(c*a+S)/c
10 B'aCb ∩ C'aBc (-a^8+4*(b^2+c^2)*a^6+32*(b+c)*S*a*b^2*c^2-2*(3*b^4+4*b^2*c^2+3*c^4)*a^4+4*(b^6+c^6+(b^2+16*b*c+c^2)*b^2*c^2)*a^2-(b^4-c^4)^2)/(4*a*S) :

(4*S*b*c+a^4-2*(b^2+c^2)*a^2+8*b^2*c*a+(b^2-c^2)^2)*c :

(4*S*b*c+a^4-2*(b^2+c^2)*a^2+8*b*c^2*a+(b^2-c^2)^2)*b
11 midpoint of A'bA'c 0 :

1/(b*((c+R)*b+3*R*c+4*R^2)) :

1/(c*((c+3*R)*b+R*c+4*R^2))
12 B"cA"c ∩ C"bA"b (-3*a^4+2*(3*b^2+2*b*c+3*c^2)*a^2+8*(b+c)*S*a-3*(b^2-c^2)^2)/(4*a*(b*c+2*S)) :

-(b*a+S)/b :

-(c*a+S)/c
13 A"bB"a ∩ A"cC"a (-3*a^4+2*(3*b^2+2*b*c+3*c^2)*a^2+8*(b+c)*S*a-3*(b^2-c^2)^2)/(4*S*a) :

-(b*a+S)/b :

-(c*a+S)/c
14 A"bB"c ∩ C"bA"c (-3*a^4+2*(3*b^2+2*b*c+3*c^2)*a^2+8*(b+c)*S*a-3*(b^2-c^2)^2)*S/(4*a*(b*c+2*S)) :

-(c*a+S)*(b*a+2*S)/b :

-(b*a+S)*(c*a+2*S)/c
15 B"aA"c ∩ C"aA"b (-3*a^4+2*(3*b^2+2*b*c+3*c^2)*a^2+8*(b+c)*S*a-3*(b^2-c^2)^2)/(4*a*S^2) :

-(c*a+S)/(c*a+2*S)/b :

-(b*a+S)/(b*a+2*S)/c
16 BB"c ∩ CC"b S/(a*(b*c+2*S)) :

-1/b :

-1/c
17 AbB"c ∩ AcC"b (a^4-2*(b-c)^2*a^2+(b^2-c^2)^2)*S/(4*a^2(*b*c+2*S)) :

-c*a-S :

-b*a-S
18 A'bB"c ∩ A'cC"b (2*a*b*c+(b+c)*S)*S/(b*c+2*S) :

-(c*a+S)*(2*b*a+S)/b :

-(b*a+S)*(2*c*a+S)/c
19 BC13 ∩ CB13 4*S/a :

-(8*(a+c)*S*b-3*a^4+6*(b^2+c^2)*a^2+4*b^2*c*a-3*(b^2-c^2)^2)/(b*(b*a+S)) :

-(8*(a+b)*S*c-3*a^4+6*(b^2+c^2)*a^2+4*b*c^2*a-3*(b^2-c^2)^2)/(c*(c*a+S))
20 AbC13 ∩ AcB13 4*(a^4-2*(b-c)^2*a^2+(b^2-c^2)^2)*S*(b*a+S)*(c*a+S)/a :

-((a+b+c)*(a^7-(b+c)*a^6-(15*b^2+22*b*c+3*c^2)*a^5+(15*b^3+3*c^3+b*c*(21*b+25*c))*a^4+(b+c)*(15*b^3+3*c^3-b*c*(3*b-17*c))*a^3-(15*b^3+3*c^3+b*c*(25*b+29*c))*(b-c)^2*a^2-(b^2-c^2)*(b-c)*(b^3+c^3+b*c*(15*b-c))*a+(b^2-c^2)^3*(b-c))*S+b*(5*a^9+c*a^8-4*(5*b^2+4*b*c+8*c^2)*a^7-4*(b+c)*(3*b+c)*c*a^6+2*(15*b^4+27*c^4+2*b*c*(8*b^2+11*b*c+8*c^2))*a^5+2*(b+c)*(11*b^3+3*c^3+b*c*(5*b+13*c))*c*a^4-4*(b^2-c^2)^2*(5*b^2+4*b*c+8*c^2)*a^3-4*(b^2-c^2)^2*(b+c)*(3*b+c)*c*a^2+5*(b^2-c^2)^4*a+(b^2-c^2)^4*c))/b :

-((a+b+c)*(a^7-(b+c)*a^6-(3*b^2+22*b*c+15*c^2)*a^5+(3*b^3+15*c^3+b*c*(25*b+21*c))*a^4+(b+c)*(3*b^3+15*c^3+b*c*(17*b-3*c))*a^3-(3*b^3+15*c^3+b*c*(29*b+25*c))*(b-c)^2*a^2-(b^2-c^2)*(b-c)*(b^3+c^3-b*c*(b-15*c))*a+(b^2-c^2)^3*(b-c))*S+c*(5*a^9+b*a^8-4*(8*b^2+4*b*c+5*c^2)*a^7-4*(b+3*c)*(b+c)*b*a^6+2*(27*b^4+15*c^4+2*b*c*(8*b^2+11*b*c+8*c^2))*a^5+2*(b+c)*(3*b^3+11*c^3+b*c*(13*b+5*c))*b*a^4-4*(b^2-c^2)^2*(8*b^2+4*b*c+5*c^2)*a^3-4*(b^2-c^2)^2*(b+c)*(b+3*c)*b*a^2+5*(b^2-c^2)^4*a+(b^2-c^2)^4*b))/c
21 A'bC13 ∩ A'cB13 16*(2*a*b*c+(b+c)*S)*S*(b*a+S)*(c*a+S) :

-((4*a^8-4*(b+c)*(9*b+7*c)*a^6-4*(7*b+4*c)*b*c*a^5+16*(b+c)*(4*b^3+3*c^3+(4*b+5*c)*b*c)*a^4+8*(7*b^3+4*c^3+(12*b+17*c)*b*c)*b*c*a^3-4*(b^2-c^2)^2*(b+c)*(9*b+7*c)*a^2-4*(b^2-c^2)^2*(7*b+4*c)*b*c*a+4*(b^2-c^2)^4)*S+2*(5*b+3*c)*a^9+3*b*c*a^8-8*(5*b^3+3*c^3+b*c*(7*b+9*c))*a^7-4*(6*b^2+10*b*c+5*c^2)*b*c*a^6+4*(15*b^5+9*c^5+(25*b^3+31*c^3+2*b*c*(13*b+11*c))*b*c)*a^5+2*(b+c)*(21*b^3+17*c^3+b*c*(19*b+23*c))*b*c*a^4-8*(b^2-c^2)^2*(5*b^3+3*c^3+b*c*(7*b+9*c))*a^3-4*(b^2-c^2)^2*(6*b^2+10*b*c+5*c^2)*b*c*a^2+2*(b^2-c^2)^4*(5*b+3*c)*a+3*(b^2-c^2)^4*b*c)/b :

-((4*a^8-4*(7*b+9*c)*(b+c)*a^6-4*(4*b+7*c)*b*c*a^5+16*(b+c)*(3*b^3+4*c^3+(5*b+4*c)*b*c)*a^4+8*(4*b^3+7*c^3+(17*b+12*c)*b*c)*b*c*a^3-4*(b^2-c^2)^2*(b+c)*(7*b+9*c)*a^2-4*(b^2-c^2)^2*(4*b+7*c)*b*c*a+4*(b^2-c^2)^4)*S+2*(3*b+5*c)*a^9+3*b*c*a^8-8*(3*b^3+5*c^3+b*c*(9*b+7*c))*a^7-4*(5*b^2+10*b*c+6*c^2)*b*c*a^6+4*(9*b^5+15*c^5+(31*b^3+25*c^3+2*b*c*(11*b+13*c))*b*c)*a^5+2*(b+c)*(17*b^3+21*c^3+b*c*(23*b+19*c))*b*c*a^4-8*(b^2-c^2)^2*(3*b^3+5*c^3+b*c*(9*b+7*c))*a^3-4*(b^2-c^2)^2*(5*b^2+10*b*c+6*c^2)*b*c*a^2+2*(b^2-c^2)^4*(3*b+5*c)*a+3*(b^2-c^2)^4*b*c)/c
22 A"bC13 ∩ A"cB13 4*(-3*a^4+2*(3*b^2+2*b*c+3*c^2)*a^2+8*(b+c)*S*a-3*(b^2-c^2)^2)*S*(b*a+S)*(c*a+S)/a :

-((5*a^8-4*(6*b^2+14*b*c+11*c^2)*a^6-8*(b+4*c)*b*c*a^5+2*(b+c)*(19*b^3+39*c^3+b*c*(37*b+17*c))*a^4+16*(b^3+4*c^3+b*c*(6*b+5*c))*b*c*a^3-4*(b^2-c^2)^2*(6*b^2+14*b*c+11*c^2)*a^2-8*(b^2-c^2)^2*(b+4*c)*b*c*a+5*(b^2-c^2)^4)*S+(5*b+12*c)*a^9+3*b*c*a^8-4*(5*b^3+12*c^3+4*b*c*(4*b+3*c))*a^7-4*(3*b^2+8*b*c+7*c^2)*b*c*a^6+2*(15*b^5+36*c^5+(52*b^3+43*c^3+2*b*c*(19*b+20*c))*b*c)*a^5+2*(b+c)*(9*b^3+25*c^3+b*c*(23*b+7*c))*b*c*a^4-4*(b^2-c^2)^2*(5*b^3+12*c^3+4*b*c*(4*b+3*c))*a^3-4*(b^2-c^2)^2*(3*b^2+8*b*c+7*c^2)*b*c*a^2+(5*b+12*c)*(b^2-c^2)^4*a+3*(b^2-c^2)^4*b*c)/b :

-((5*a^8-4*(11*b^2+14*b*c+6*c^2)*a^6-8*(4*b+c)*b*c*a^5+2*(b+c)*(39*b^3+19*c^3+b*c*(17*b+37*c))*a^4+16*(4*b^3+c^3+b*c*(5*b+6*c))*b*c*a^3-4*(b^2-c^2)^2*(11*b^2+14*b*c+6*c^2)*a^2-8*(b^2-c^2)^2*(4*b+c)*b*c*a+5*(b^2-c^2)^4)*S+(12*b+5*c)*a^9+3*b*c*a^8-4*(12*b^3+5*c^3+4*b*c*(3*b+4*c))*a^7-4*(7*b^2+8*b*c+3*c^2)*b*c*a^6+2*(36*b^5+15*c^5+(43*b^3+52*c^3+2*b*c*(20*b+19*c))*b*c)*a^5+2*(b+c)*(25*b^3+9*c^3+b*c*(7*b+23*c))*b*c*a^4-4*(b^2-c^2)^2*(12*b^3+5*c^3+4*b*c*(3*b+4*c))*a^3-4*(b^2-c^2)^2*(7*b^2+8*b*c+3*c^2)*b*c*a^2+(12*b+5*c)*(b^2-c^2)^4*a+3*(b^2-c^2)^4*b*c)/c
23 C"aB13 ∩ B"aC13 (16*(a+b+c)*(7*(b+c)*a^8-(7*b^2+6*b*c+7*c^2)*a^7-3*(b+c)*(7*b^2+2*b*c+7*c^2)*a^6+(21*b^4+21*c^4+2*(2*b^2+3*b*c+2*c^2)*b*c)*a^5+(b+c)*(21*b^4+21*c^4+4*(5*b^2+2*b*c+5*c^2)*b*c)*a^4-(21*b^6+21*c^6-(10*b^4+10*c^4+(27*b^2+40*b*c+27*c^2)*b*c)*b*c)*a^3-(b^2-c^2)*(b-c)*(7*b^4+7*c^4+36*(b^2+b*c+c^2)*b*c)*a^2+(7*b^4+7*c^4-4*(2*b^2+3*b*c+2*c^2)*b*c)*(b^2-c^2)^2*a+8*(b^2-c^2)^3*(b-c)*b*c)*S-21*a^12+2*(81*b^2+74*b*c+81*c^2)*a^10+160*(b+c)*b*c*a^9-(459*b^4+459*c^4+2*(296*b^2+341*b*c+296*c^2)*b*c)*a^8-32*(b+c)*(20*b^2+7*b*c+20*c^2)*b*c*a^7+4*(159*b^6+159*c^6+(222*b^4+222*c^4+11*(11*b^2+8*b*c+11*c^2)*b*c)*b*c)*a^6+64*(b+c)*(15*b^4+15*c^4+(7*b^2+10*b*c+7*c^2)*b*c)*b*c*a^5-(459*b^8+459*c^8+2*(296*b^6+296*c^6-(302*b^4+302*c^4+(536*b^2+527*b*c+536*c^2)*b*c)*b*c)*b*c)*a^4-32*(b^2-c^2)^2*(b+c)*(20*b^2+7*b*c+20*c^2)*b*c*a^3+2*(b^2-c^2)^2*(81*b^6+81*c^6+(74*b^4+74*c^4-(209*b^2+268*b*c+209*c^2)*b*c)*b*c)*a^2+160*(b^2-c^2)^4*(b+c)*b*c*a+(b^2-c^2)^4*(-21*b^4+90*b^2*c^2-21*c^4))/(16*a*S^2*(b*c+S)) :

((3*a^4-2*(3*b^2+4*b*c+3*c^2)*a^2-4*(b+2*c)*b*c*a+3*(b^2-c^2)^2)*S+(2*b+3*c)*a^5+2*b*c*a^4-2*(2*b+3*c)*(b^2+c^2)*a^3-4*(b^2+b*c+c^2)*b*c*a^2+(2*b+3*c)*(b^2-c^2)^2*a+2*(b^2-c^2)^2*b*c)/b :

((3*a^4-2*(3*b^2+4*b*c+3*c^2)*a^2-4*(2*b+c)*b*c*a+3*(b^2-c^2)^2)*S+(3*b+2*c)*a^5+2*b*c*a^4-2*(3*b+2*c)*(b^2+c^2)*a^3-4*(b^2+b*c+c^2)*b*c*a^2+(3*b+2*c)*(b^2-c^2)^2*a+2*(b^2-c^2)^2*b*c)/c
24 A14A16 ∩ BC 0 :

1/((c*a+S)*b) :

1/((b*a+S)*c)
25 BC15 ∩ CB15 4*S^2*(b*c+S)/((b*c+2*S)*a) :

-(8*(a+c)*S*b-3*a^4+6*(b^2+c^2)*a^2+4*b^2*c*a-3*(b^2-c^2)^2)/b :

-(8*(a+b)*S*c-3*a^4+6*(b^2+c^2)*a^2+4*b*c^2*a-3*(b^2-c^2)^2)/c
26 AbC15 ∩ AcB15 8*(a^4-2*(b-c)^2*a^2+(b^2-c^2)^2)*S^2*(b*a+2*S)*(b*c+S)*(c*a+2*S)/((b*c+2*S)*a) :

-(2*a*((15*b+2*c)*a^8-4*(15*b^3+2*c^3+b*c*(24*b+25*c))*a^6-16*(b+4*c)*b^2*c*a^5+2*(45*b^5+6*c^5+(94*b^3+85*c^3+2*b*c*(43*b+46*c))*b*c)*a^4+16*(2*b^3+8*c^3+b*c*(10*b+7*c))*b^2*c*a^3-4*(b^2-c^2)^2*(15*b^3+2*c^3+b*c*(24*b+25*c))*a^2-16*(b^2-c^2)^2*(b+4*c)*b^2*c*a+(15*b+2*c)*(b^2-c^2)^4)*S-a^12+2*(11*b^2+28*b*c+3*c^2)*a^10+16*b^2*c*a^9-(79*b^4+15*c^4+2*b*c*(128*b^2+77*b*c+112*c^2))*a^8-16*(4*b^2+6*b*c+7*c^2)*b^2*c*a^7+4*(29*b^6+5*c^6+(100*b^4+84*c^4+b*c*(55*b^2+72*b*c+63*c^2))*b*c)*a^6+96*(b^4+2*c^4+2*(b^2+b*c+c^2)*b*c)*b^2*c*a^5-(b^2-c^2)^2*(79*b^4+15*c^4+2*b*c*(128*b^2+77*b*c+112*c^2))*a^4-16*(b^2-c^2)^2*(4*b^2+6*b*c+7*c^2)*b^2*c*a^3+2*(b^2-c^2)^4*(11*b^2+28*b*c+3*c^2)*a^2+16*(b^2-c^2)^4*b^2*c*a-(b^2-c^2)^6)/b :

-(2*a*((2*b+15*c)*a^8-4*(2*b^3+15*c^3+b*c*(25*b+24*c))*a^6-16*(4*b+c)*b*c^2*a^5+2*(6*b^5+45*c^5+(85*b^3+94*c^3+2*b*c*(46*b+43*c))*b*c)*a^4+16*(8*b^3+2*c^3+b*c*(7*b+10*c))*b*c^2*a^3-4*(b^2-c^2)^2*(2*b^3+15*c^3+b*c*(25*b+24*c))*a^2-16*(b^2-c^2)^2*(4*b+c)*b*c^2*a+(2*b+15*c)*(b^2-c^2)^4)*S-a^12+2*(3*b^2+28*b*c+11*c^2)*a^10+16*b*c^2*a^9-(15*b^4+79*c^4+2*b*c*(112*b^2+77*b*c+128*c^2))*a^8-16*(7*b^2+6*b*c+4*c^2)*b*c^2*a^7+4*(5*b^6+29*c^6+(84*b^4+100*c^4+b*c*(63*b^2+72*b*c+55*c^2))*b*c)*a^6+96*(2*b^4+c^4+2*(b^2+b*c+c^2)*b*c)*b*c^2*a^5-(b^2-c^2)^2*(15*b^4+79*c^4+2*b*c*(112*b^2+77*b*c+128*c^2))*a^4-16*(b^2-c^2)^2*(7*b^2+6*b*c+4*c^2)*b*c^2*a^3+2*(b^2-c^2)^4*(3*b^2+28*b*c+11*c^2)*a^2+16*(b^2-c^2)^4*b*c^2*a-(b^2-c^2)^6)/c
27 A'bC15 ∩ A'cB15 32*(2*a*b*c+(b+c)*S)*S^2*(b*a+2*S)*(b*c+S)*(c*a+2*S)/(b*c+2*S) :

-(((75*b+47*c)*a^9+16*b*c*a^8-4*(83*b^3+47*c^3+b*c*(143*b+123*c))*a^7-8*(21*b^2+40*b*c+13*c^2)*b*c*a^6+2*(257*b^5+141*c^5+(525*b^3+417*c^3+2*b*c*(219*b+239*c))*b*c)*a^5+16*(19*b^4+11*c^4+2*b*c*(25*b^2+21*b*c+20*c^2))*b*c*a^4-4*(b^2-c^2)^2*(83*b^3+47*c^3+b*c*(143*b+123*c))*a^3-8*(b^2-c^2)^2*(21*b^2+40*b*c+13*c^2)*b*c*a^2+(75*b+47*c)*(b^2-c^2)^4*a+16*(b^2-c^2)^4*b*c)*S-7*a^12+2*(45*b^2+67*b*c+29*c^2)*a^10+4*(11*b+7*c)*b*c*a^9-(297*b^4+169*c^4+2*b*c*(308*b^2+263*b*c+268*c^2))*a^8-4*(48*b^3+28*c^3+b*c*(81*b+73*c))*b*c*a^7+4*(107*b^6+59*c^6+(241*b^4+201*c^4+(165*b^2+174*b*c+181*c^2)*b*c)*b*c)*a^6+8*(b+c)*(37*b^4+21*c^4+b*c*(37*b^2+26*b*c+41*c^2))*b*c*a^5-(b^2-c^2)^2*(297*b^4+169*c^4+2*b*c*(308*b^2+263*b*c+268*c^2))*a^4-4*(b^2-c^2)^2*(48*b^3+28*c^3+b*c*(81*b+73*c))*b*c*a^3+2*(b^2-c^2)^4*(45*b^2+67*b*c+29*c^2)*a^2+4*(b^2-c^2)^4*(11*b+7*c)*b*c*a-7*(b^2-c^2)^6)/b :

-(((47*b+75*c)*a^9+16*b*c*a^8-4*(47*b^3+83*c^3+(123*b+143*c)*b*c)*a^7-8*(13*b^2+40*b*c+21*c^2)*b*c*a^6+2*(141*b^5+257*c^5+(417*b^3+525*c^3+2*(239*b+219*c)*b*c)*b*c)*a^5+16*(11*b^4+19*c^4+2*(20*b^2+21*b*c+25*c^2)*b*c)*b*c*a^4-4*(b^2-c^2)^2*(47*b^3+83*c^3+(123*b+143*c)*b*c)*a^3-8*(b^2-c^2)^2*(13*b^2+40*b*c+21*c^2)*b*c*a^2+(47*b+75*c)*(b^2-c^2)^4*a+16*(b^2-c^2)^4*b*c)*S-7*a^12+2*(29*b^2+67*b*c+45*c^2)*a^10+4*(7*b+11*c)*b*c*a^9-(169*b^4+297*c^4+2*b*c*(268*b^2+263*b*c+308*c^2))*a^8-4*(28*b^3+48*c^3+b*c*(73*b+81*c))*b*c*a^7+4*(59*b^6+107*c^6+(201*b^4+241*c^4+b*c*(181*b^2+174*b*c+165*c^2))*b*c)*a^6+8*(b+c)*(21*b^4+37*c^4+b*c*(41*b^2+26*b*c+37*c^2))*b*c*a^5-(b^2-c^2)^2*(169*b^4+297*c^4+2*b*c*(268*b^2+263*b*c+308*c^2))*a^4-4*(b^2-c^2)^2*(28*b^3+48*c^3+b*c*(73*b+81*c))*b*c*a^3+2*(b^2-c^2)^4*(29*b^2+67*b*c+45*c^2)*a^2+4*(b^2-c^2)^4*(7*b+11*c)*b*c*a-7*(b^2-c^2)^6)/c
28 A"bC15 ∩ A"cB15 4*(-3*a^4+2*(3*b^2+2*b*c+3*c^2)*a^2+8*(b+c)*S*a-3*(b^2-c^2)^2)*(b*c+S)*S^2/(b*c+2*S)/a :

-((11*a^8-4*(19*b^2+20*b*c+13*c^2)*a^6-16*(3*b+2*c)*b*c*a^5+2*(65*b^4+41*c^4+2*b*c*(40*b^2+39*b*c+40*c^2))*a^4+16*(6*b^3+4*c^3+b*c*(8*b+9*c))*b*c*a^3-4*(b^2-c^2)^2*(19*b^2+20*b*c+13*c^2)*a^2-16*(b^2-c^2)^2*(3*b+2*c)*b*c*a+11*(b^2-c^2)^4)*S+10*(2*b+c)*a^9+8*b*c*a^8-4*(20*b^3+10*c^3+b*c*(18*b+25*c))*a^7-8*(6*b^2+6*b*c+5*c^2)*b*c*a^6+4*(30*b^5+15*c^5+(31*b^3+40*c^3+2*b*c*(15*b+13*c))*b*c)*a^5+16*(5*b^4+4*c^4+6*(b^2+b*c+c^2)*b*c)*b*c*a^4-4*(b^2-c^2)^2*(20*b^3+10*c^3+b*c*(18*b+25*c))*a^3-8*(b^2-c^2)^2*(6*b^2+6*b*c+5*c^2)*b*c*a^2+10*(b^2-c^2)^4*(2*b+c)*a+8*(b^2-c^2)^4*b*c)/(c*a+2*S)/b :

-((11*a^8-4*(13*b^2+20*b*c+19*c^2)*a^6-16*(2*b+3*c)*b*c*a^5+2*(41*b^4+65*c^4+2*b*c*(40*b^2+39*b*c+40*c^2))*a^4+16*(4*b^3+6*c^3+b*c*(9*b+8*c))*b*c*a^3-4*(b^2-c^2)^2*(13*b^2+20*b*c+19*c^2)*a^2-16*(b^2-c^2)^2*(2*b+3*c)*b*c*a+11*(b^2-c^2)^4)*S+10*(b+2*c)*a^9+8*b*c*a^8-4*(10*b^3+20*c^3+b*c*(25*b+18*c))*a^7-8*(5*b^2+6*b*c+6*c^2)*b*c*a^6+4*(15*b^5+30*c^5+(40*b^3+31*c^3+2*b*c*(13*b+15*c))*b*c)*a^5+16*(4*b^4+5*c^4+6*(b^2+b*c+c^2)*b*c)*b*c*a^4-4*(b^2-c^2)^2*(10*b^3+20*c^3+b*c*(25*b+18*c))*a^3-8*(b^2-c^2)^2*(5*b^2+6*b*c+6*c^2)*b*c*a^2+10*(b^2-c^2)^4*(b+2*c)*a+8*(b^2-c^2)^4*b*c)/(b*a+2*S)/c
29 B"aC15 ∩ C"aB15 (-117*a^16+1372*(b+c)*b*c*a^13+8*(141*b^2+92*b*c+141*c^2)*a^14-8*(b+c)*(1029*b^2+232*b*c+1029*c^2)*b*c*a^11-12*(369*b^4+369*c^4+2*(184*b^2+259*b*c+184*c^2)*b*c)*a^12+4*(b+c)*(5145*b^4+5145*c^4+2*(928*b^2+2863*b*c+928*c^2)*b*c)*b*c*a^9+8*(1179*b^6+1179*c^6+(1380*b^4+1380*c^4+(1037*b^2+1004*b*c+1037*c^2)*b*c)*b*c)*a^10-16*(b+c)*(1715*b^6+1715*c^6+(696*b^4+696*c^4+(581*b^2+340*b*c+581*c^2)*b*c)*b*c)*b*c*a^7-2*(6015*b^8+6015*c^8+2*(3680*b^6+3680*c^6-(2094*b^4+2094*c^4+(3008*b^2+5155*b*c+3008*c^2)*b*c)*b*c)*b*c)*a^8+4*(b+c)*(5145*b^8+5145*c^8+2*(928*b^6+928*c^6-(3402*b^4+3402*c^4+(1424*b^2+1925*b*c+1424*c^2)*b*c)*b*c)*b*c)*b*c*a^5+8*(1179*b^10+1179*c^10+(1380*b^8+1380*c^8-(3953*b^6+3953*c^6+2*(2508*b^4+2508*c^4+(2117*b^2+1516*b*c+2117*c^2)*b*c)*b*c)*b*c)*b*c)*a^6-8*(b^2-c^2)^2*(b+c)*(1029*b^6+1029*c^6+(232*b^4+232*c^4-(1925*b^2+712*b*c+1925*c^2)*b*c)*b*c)*b*c*a^3-4*(1107*b^12+1107*c^12+(1104*b^10+1104*c^10-(7802*b^8+7802*c^8+(8528*b^6+8528*c^6-(3709*b^4+3709*c^4+4*(1984*b^2+1893*b*c+1984*c^2)*b*c)*b*c)*b*c)*b*c)*b*c)*a^4+28*(b^2-c^2)^4*(b+c)*(7*b^2-8*b*c-7*c^2)*(7*b^2+8*b*c-7*c^2)*b*c*a+8*(b^2-c^2)^2*(141*b^10+141*c^10+(92*b^8+92*c^8-(1359*b^6+1359*c^6+2*(510*b^4+510*c^4-(689*b^2+992*b*c+689*c^2)*b*c)*b*c)*b*c)*b*c)*a^2+(600*(b+c)*a^13+1128*b*c*a^12-16*(b+c)*(225*b^2+56*b*c+225*c^2)*a^11-16*(527*b^2+400*b*c+527*c^2)*b*c*a^10+8*(b+c)*(1125*b^4+1125*c^4+2*(224*b^2+381*b*c+224*c^2)*b*c)*a^9+8*(2947*b^4+2947*c^4+50*(64*b^2+85*b*c+64*c^2)*b*c)*b*c*a^8-32*(b+c)*(375*b^6+375*c^6+(168*b^4+168*c^4-(363*b^2+68*b*c+363*c^2)*b*c)*b*c)*a^7-32*(1017*b^6+1017*c^6+(1200*b^4+1200*c^4+(803*b^2+564*b*c+803*c^2)*b*c)*b*c)*b*c*a^6+8*(b+c)*(1125*b^8+1125*c^8+2*(224*b^6+224*c^6-(2214*b^4+2214*c^4+(944*b^2+1337*b*c+944*c^2)*b*c)*b*c)*b*c)*a^5+8*(2947*b^8+2947*c^8+2*(1600*b^6+1600*c^6-(1582*b^4+1582*c^4+3*(848*b^2+845*b*c+848*c^2)*b*c)*b*c)*b*c)*b*c*a^4-16*(b+c)*(225*b^10+225*c^10+(56*b^8+56*c^8-(1851*b^6+1851*c^6+2*(292*b^4+292*c^4-(877*b^2+560*b*c+877*c^2)*b*c)*b*c)*b*c)*b*c)*a^3-16*(b^2-c^2)^2*(527*b^6+527*c^6+(400*b^4+400*c^4-(1063*b^2+1272*b*c+1063*c^2)*b*c)*b*c)*b*c*a^2+8*(b^2-c^2)^2*(b+c)*(75*b^8+75*c^8-2*(444*b^4-877*b^2*c^2+444*c^4)*b^2*c^2)*a+8*(b^2-c^2)^4*(141*b^4-490*b^2*c^2+141*c^4)*b*c)*S-(b^2-c^2)^4*(117*b^8+117*c^8-2*(746*b^4-1503*b^2*c^2+746*c^4)*b^2*c^2))/S^3/(b*c+2*S)/(-9*a^8+28*(b+c)*b*c*a^5+4*(9*b^2+4*b*c+9*c^2)*a^6-56*(b+c)*(b^2+c^2)*b*c*a^3-2*(27*b^4+27*c^4+2*(8*b^2+5*b*c+8*c^2)*b*c)*a^4+28*(b^2-c^2)^2*(b+c)*b*c*a+4*(9*b^6+9*c^6+(4*b^4+4*c^4-(17*b^2+12*b*c+17*c^2)*b*c)*b*c)*a^2+(24*(b+c)*a^5+48*b*c*a^4-48*(b+c)*(b^2+c^2)*a^3-32*(3*b^2+2*b*c+3*c^2)*b*c*a^2+8*(b+c)*(b^2-3*c^2)*(3*b^2-c^2)*a+48*(b^2-c^2)^2*b*c)*S-(b^2-c^2)^2*(3*b^2-4*b*c-3*c^2)*(3*b^2+4*b*c-3*c^2))/(16*a) :

-(c*a+S)/(c*a+2*S)/b :

-(b*a+S)/(b*a+2*S)/c
30 midpoint of C"bB"c -1/(a+R) :

1/b :

1/c
31 midpoint of B"aC"a (2*(2*c+3*R)*b+6*R*c+8*R^2)/a :

-2*R-c :

-b-2*R

References:
  (1): Preamble just before X(37994)
  (2): Preamble just before X(38487)
  (3): Preamble just before X(39311)

CÚsar Eliud Lozada, August 21, 2021 - Corrected: February 20, 2024