One direct way to derive the second equation (27) is a follows. We use Mathematica with the code:

<< PolyhedronOperations`

PolyhedronData["Icosahedron", "Edges"]

Then we obtain a list of the edges of the icosahedron as line primitives:

GraphicsComplex[{{0, 0, -(5/Sqrt[50 - 10 Sqrt[5]])}, {0, 0, 5/Sqrt[

50 - 10 Sqrt[5]]}, {-Sqrt[(2/(5 - Sqrt[5]))],

0, -(1/Sqrt[10 - 2 Sqrt[5]])}, {Sqrt[2/(5 - Sqrt[5])], 0, 1/Sqrt[

10 - 2 Sqrt[5]]}, {(1 + Sqrt[5])/(

2 Sqrt[10 - 2 Sqrt[5]]), -(1/2), -(1/Sqrt[10 - 2 Sqrt[5]])}, {(

1 + Sqrt[5])/(2 Sqrt[10 - 2 Sqrt[5]]), 1/

2, -(1/Sqrt[10 - 2 Sqrt[5]])}, {-((1 + Sqrt[5])/(

2 Sqrt[10 - 2 Sqrt[5]])), -(1/2), 1/Sqrt[

10 - 2 Sqrt[5]]}, {-((1 + Sqrt[5])/(2 Sqrt[10 - 2 Sqrt[5]])), 1/2,

1/Sqrt[10 - 2 Sqrt[5]]}, {-((-1 + Sqrt[5])/(

2 Sqrt[10 - 2 Sqrt[5]])), -(1/2) Sqrt[(5 + Sqrt[5])/(

5 - Sqrt[5])], -(1/Sqrt[10 - 2 Sqrt[5]])}, {-((-1 + Sqrt[5])/(

2 Sqrt[10 - 2 Sqrt[5]])),

1/2 Sqrt[(5 + Sqrt[5])/(5 - Sqrt[5])], -(1/Sqrt[

10 - 2 Sqrt[5]])}, {(-1 + Sqrt[5])/(

2 Sqrt[10 - 2 Sqrt[5]]), -(1/2) Sqrt[(5 + Sqrt[5])/(5 - Sqrt[5])],

1/Sqrt[10 - 2 Sqrt[5]]}, {(-1 + Sqrt[5])/(2 Sqrt[10 - 2 Sqrt[5]]),

1/2 Sqrt[(5 + Sqrt[5])/(5 - Sqrt[5])], 1/Sqrt[10 - 2 Sqrt[5]]}},

Line[{{1, 3}, {1, 5}, {1, 6}, {1, 9}, {1, 10}, {2, 4}, {2, 7}, {2,

8}, {2, 11}, {2, 12}, {3, 7}, {3, 8}, {3, 9}, {3, 10}, {4, 5}, {4,

6}, {4, 11}, {4, 12}, {5, 6}, {5, 9}, {5, 11}, {6, 10}, {6,

12}, {7, 8}, {7, 9}, {7, 11}, {8, 10}, {8, 12}, {9, 11}, {10,

12}}]]

From such list we obtain the coordinates (normalized) of the edge midpoints of the icosahedron:

[[-1/10*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2), 0, -1/20*5^(1/2)*(10-2*5^(1/2))^(1/2)-1/4*(10-2*5^(1/2))^(1/2)], [1/8*(10-2*5^(1/2))^(1/2)+1/40*5^(1/2)*(10-2*5^(1/2))^(1/2), 1/4-1/4*5^(1/2), -1/20*5^(1/2)*(10-2*5^(1/2))^(1/2)-1/4*(10-2*5^(1/2))^(1/2)], [1/8*(10-2*5^(1/2))^(1/2)+1/40*5^(1/2)*(10-2*5^(1/2))^(1/2), -1/4+1/4*5^(1/2), -1/20*5^(1/2)*(10-2*5^(1/2))^(1/2)-1/4*(10-2*5^(1/2))^(1/2)], [1/40*5^(1/2)*(10-2*5^(1/2))^(1/2)-1/8*(10-2*5^(1/2))^(1/2), -1/2, -1/20*5^(1/2)*(10-2*5^(1/2))^(1/2)-1/4*(10-2*5^(1/2))^(1/2)], [1/40*5^(1/2)*(10-2*5^(1/2))^(1/2)-1/8*(10-2*5^(1/2))^(1/2), 1/2, -1/20*5^(1/2)*(10-2*5^(1/2))^(1/2)-1/4*(10-2*5^(1/2))^(1/2)], [1/10*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2), 0, 1/20*5^(1/2)*(10-2*5^(1/2))^(1/2)+1/4*(10-2*5^(1/2))^(1/2)], [-1/40*5^(1/2)*(10-2*5^(1/2))^(1/2)-1/8*(10-2*5^(1/2))^(1/2), 1/4-1/4*5^(1/2), 1/20*5^(1/2)*(10-2*5^(1/2))^(1/2)+1/4*(10-2*5^(1/2))^(1/2)], [-1/40*5^(1/2)*(10-2*5^(1/2))^(1/2)-1/8*(10-2*5^(1/2))^(1/2), -1/4+1/4*5^(1/2), 1/20*5^(1/2)*(10-2*5^(1/2))^(1/2)+1/4*(10-2*5^(1/2))^(1/2)], [-1/40*5^(1/2)*(10-2*5^(1/2))^(1/2)+1/8*(10-2*5^(1/2))^(1/2), -1/2, 1/20*5^(1/2)*(10-2*5^(1/2))^(1/2)+1/4*(10-2*5^(1/2))^(1/2)], [-1/40*5^(1/2)*(10-2*5^(1/2))^(1/2)+1/8*(10-2*5^(1/2))^(1/2), 1/2, 1/20*5^(1/2)*(10-2*5^(1/2))^(1/2)+1/4*(10-2*5^(1/2))^(1/2)], [-1/8*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)-1/8*(5-5^(1/2))^(1/2)*2^(1/2), 1/4-1/4*5^(1/2), 0], [-1/8*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)-1/8*(5-5^(1/2))^(1/2)*2^(1/2), -1/4+1/4*5^(1/2), 0], [-3/40*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)-1/8*(5-5^(1/2))^(1/2)*2^(1/2), -1/2, -1/10*5^(1/2)*(10-2*5^(1/2))^(1/2)], [-3/40*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)-1/8*(5-5^(1/2))^(1/2)*2^(1/2), 1/2, -1/10*5^(1/2)*(10-2*5^(1/2))^(1/2)], [1/8*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)+1/8*(5-5^(1/2))^(1/2)*2^(1/2), 1/4-1/4*5^(1/2), 0], [1/8*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)+1/8*(5-5^(1/2))^(1/2)*2^(1/2), -1/4+1/4*5^(1/2), 0], [3/40*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)+1/8*(5-5^(1/2))^(1/2)*2^(1/2), -1/2, 1/10*5^(1/2)*(10-2*5^(1/2))^(1/2)], [3/40*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)+1/8*(5-5^(1/2))^(1/2)*2^(1/2), 1/2, 1/10*5^(1/2)*(10-2*5^(1/2))^(1/2)], [1/20*5^(1/2)*(10-2*5^(1/2))^(1/2)+1/4*(10-2*5^(1/2))^(1/2), 0, -1/10*5^(1/2)*(10-2*5^(1/2))^(1/2)], [1/20*5^(1/2)*(10-2*5^(1/2))^(1/2), -1/4-1/4*5^(1/2), -1/10*5^(1/2)*(10-2*5^(1/2))^(1/2)], [1/4*(10-2*5^(1/2))^(1/2), -1/4-1/4*5^(1/2), 0], [1/20*5^(1/2)*(10-2*5^(1/2))^(1/2), 1/4+1/4*5^(1/2), -1/10*5^(1/2)*(10-2*5^(1/2))^(1/2)], [1/4*(10-2*5^(1/2))^(1/2), 1/4+1/4*5^(1/2), 0], [-1/20*5^(1/2)*(10-2*5^(1/2))^(1/2)-1/4*(10-2*5^(1/2))^(1/2), 0, 1/10*5^(1/2)*(10-2*5^(1/2))^(1/2)], [-1/4*(10-2*5^(1/2))^(1/2), -1/4-1/4*5^(1/2), 0], [-1/20*5^(1/2)*(10-2*5^(1/2))^(1/2), -1/4-1/4*5^(1/2), 1/10*5^(1/2)*(10-2*5^(1/2))^(1/2)], [-1/4*(10-2*5^(1/2))^(1/2), 1/4+1/4*5^(1/2), 0], [-1/20*5^(1/2)*(10-2*5^(1/2))^(1/2), 1/4+1/4*5^(1/2), 1/10*5^(1/2)*(10-2*5^(1/2))^(1/2)], [0, -1, 0], [0, 1, 0]]

From the last list we obtain the corresponding P vector given by

P := [[1, -2*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)/(20+(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)+5*(5-5^(1/2))^(1/2)*2^(1/2))], [1, -1/2*(-5*(10-2*5^(1/2))^(1/2)-5^(1/2)*(10-2*5^(1/2))^(1/2)-10*I+10*I*5^(1/2))/(20+5^(1/2)*(10-2*5^(1/2))^(1/2)+5*(10-2*5^(1/2))^(1/2))], [1, 1/2*(5*(10-2*5^(1/2))^(1/2)+5^(1/2)*(10-2*5^(1/2))^(1/2)-10*I+10*I*5^(1/2))/(20+5^(1/2)*(10-2*5^(1/2))^(1/2)+5*(10-2*5^(1/2))^(1/2))], [1, -1/2*(-5^(1/2)*(10-2*5^(1/2))^(1/2)+5*(10-2*5^(1/2))^(1/2)+20*I)/(20+5^(1/2)*(10-2*5^(1/2))^(1/2)+5*(10-2*5^(1/2))^(1/2))], [1, 1/2*(5^(1/2)*(10-2*5^(1/2))^(1/2)-5*(10-2*5^(1/2))^(1/2)+20*I)/(20+5^(1/2)*(10-2*5^(1/2))^(1/2)+5*(10-2*5^(1/2))^(1/2))], [1, -2*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)/(-20+(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)+5*(5-5^(1/2))^(1/2)*2^(1/2))], [1, 1/2*(5*(10-2*5^(1/2))^(1/2)+5^(1/2)*(10-2*5^(1/2))^(1/2)-10*I+10*I*5^(1/2))/(-20+5^(1/2)*(10-2*5^(1/2))^(1/2)+5*(10-2*5^(1/2))^(1/2))], [1, -1/2*(-5*(10-2*5^(1/2))^(1/2)-5^(1/2)*(10-2*5^(1/2))^(1/2)-10*I+10*I*5^(1/2))/(-20+5^(1/2)*(10-2*5^(1/2))^(1/2)+5*(10-2*5^(1/2))^(1/2))], [1, 1/2*(5^(1/2)*(10-2*5^(1/2))^(1/2)-5*(10-2*5^(1/2))^(1/2)+20*I)/(-20+5^(1/2)*(10-2*5^(1/2))^(1/2)+5*(10-2*5^(1/2))^(1/2))], [1, 1/2*(5^(1/2)*(10-2*5^(1/2))^(1/2)-5*(10-2*5^(1/2))^(1/2)-20*I)/(-20+5^(1/2)*(10-2*5^(1/2))^(1/2)+5*(10-2*5^(1/2))^(1/2))], [1, -1/8*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)-1/8*(5-5^(1/2))^(1/2)*2^(1/2)+1/4*I-1/4*I*5^(1/2)], [1, -1/8*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)-1/8*(5-5^(1/2))^(1/2)*2^(1/2)-1/4*I+1/4*I*5^(1/2)], [1, -1/4*(3*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)+5*(5-5^(1/2))^(1/2)*2^(1/2)+20*I)/(10+(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2))], [1, -1/4*(3*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)+5*(5-5^(1/2))^(1/2)*2^(1/2)-20*I)/(10+(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2))], [1, 1/8*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)+1/8*(5-5^(1/2))^(1/2)*2^(1/2)+1/4*I-1/4*I*5^(1/2)], [1, 1/8*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)+1/8*(5-5^(1/2))^(1/2)*2^(1/2)-1/4*I+1/4*I*5^(1/2)], [1, 1/4*(-3*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)-5*(5-5^(1/2))^(1/2)*2^(1/2)+20*I)/(-10+(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2))], [1, -1/4*(3*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)+5*(5-5^(1/2))^(1/2)*2^(1/2)+20*I)/(-10+(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2))], [1, 1/2*(10-2*5^(1/2))^(1/2)*(5+5^(1/2))/(10+5^(1/2)*(10-2*5^(1/2))^(1/2))], [1, -1/2*(-5^(1/2)*(10-2*5^(1/2))^(1/2)+5*I+5*I*5^(1/2))/(10+5^(1/2)*(10-2*5^(1/2))^(1/2))], [1, 1/4*(10-2*5^(1/2))^(1/2)-1/4*I-1/4*I*5^(1/2)], [1, 1/2*(5^(1/2)*(10-2*5^(1/2))^(1/2)+5*I+5*I*5^(1/2))/(10+5^(1/2)*(10-2*5^(1/2))^(1/2))], [1, 1/4*(10-2*5^(1/2))^(1/2)+1/4*I+1/4*I*5^(1/2)], [1, 1/2*(10-2*5^(1/2))^(1/2)*(5+5^(1/2))/(-10+5^(1/2)*(10-2*5^(1/2))^(1/2))], [1, -1/4*(10-2*5^(1/2))^(1/2)-1/4*I-1/4*I*5^(1/2)], [1, 1/2*(5^(1/2)*(10-2*5^(1/2))^(1/2)+5*I+5*I*5^(1/2))/(-10+5^(1/2)*(10-2*5^(1/2))^(1/2))], [1, -1/4*(10-2*5^(1/2))^(1/2)+1/4*I+1/4*I*5^(1/2)], [1, -1/2*(-5^(1/2)*(10-2*5^(1/2))^(1/2)+5*I+5*I*5^(1/2))/(-10+5^(1/2)*(10-2*5^(1/2))^(1/2))], [1, -I], [1, I]]

From this P and using the standard procedure we obtain the following polynomial

(alpha/beta+1/20/(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)*(20+(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)+5*(5-5^(1/2))^(1/2)*2^(1/2)))*(alpha/beta+2/(-5*(10-2*5^(1/2))^(1/2)-5^(1/2)*(10-2*5^(1/2))^(1/2)-10*I+10*I*5^(1/2))*(20+5^(1/2)*(10-2*5^(1/2))^(1/2)+5*(10-2*5^(1/2))^(1/2)))*(alpha/beta-2/(5*(10-2*5^(1/2))^(1/2)+5^(1/2)*(10-2*5^(1/2))^(1/2)-10*I+10*I*5^(1/2))*(20+5^(1/2)*(10-2*5^(1/2))^(1/2)+5*(10-2*5^(1/2))^(1/2)))*(alpha/beta+2/(-5^(1/2)*(10-2*5^(1/2))^(1/2)+5*(10-2*5^(1/2))^(1/2)+20*I)*(20+5^(1/2)*(10-2*5^(1/2))^(1/2)+5*(10-2*5^(1/2))^(1/2)))*(alpha/beta-2/(5^(1/2)*(10-2*5^(1/2))^(1/2)-5*(10-2*5^(1/2))^(1/2)+20*I)*(20+5^(1/2)*(10-2*5^(1/2))^(1/2)+5*(10-2*5^(1/2))^(1/2)))*(alpha/beta+1/20/(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)*(-20+(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)+5*(5-5^(1/2))^(1/2)*2^(1/2)))*(alpha/beta-2/(5*(10-2*5^(1/2))^(1/2)+5^(1/2)*(10-2*5^(1/2))^(1/2)-10*I+10*I*5^(1/2))*(-20+5^(1/2)*(10-2*5^(1/2))^(1/2)+5*(10-2*5^(1/2))^(1/2)))*(alpha/beta+2/(-5*(10-2*5^(1/2))^(1/2)-5^(1/2)*(10-2*5^(1/2))^(1/2)-10*I+10*I*5^(1/2))*(-20+5^(1/2)*(10-2*5^(1/2))^(1/2)+5*(10-2*5^(1/2))^(1/2)))*(alpha/beta-2/(5^(1/2)*(10-2*5^(1/2))^(1/2)-5*(10-2*5^(1/2))^(1/2)+20*I)*(-20+5^(1/2)*(10-2*5^(1/2))^(1/2)+5*(10-2*5^(1/2))^(1/2)))*(alpha/beta-2/(5^(1/2)*(10-2*5^(1/2))^(1/2)-5*(10-2*5^(1/2))^(1/2)-20*I)*(-20+5^(1/2)*(10-2*5^(1/2))^(1/2)+5*(10-2*5^(1/2))^(1/2)))*(alpha/beta-1/(-1/8*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)-1/8*(5-5^(1/2))^(1/2)*2^(1/2)+1/4*I-1/4*I*5^(1/2)))*(alpha/beta-1/(-1/8*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)-1/8*(5-5^(1/2))^(1/2)*2^(1/2)-1/4*I+1/4*I*5^(1/2)))*(alpha/beta+4/(3*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)+5*(5-5^(1/2))^(1/2)*2^(1/2)+20*I)*(10+(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)))*(alpha/beta+4/(3*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)+5*(5-5^(1/2))^(1/2)*2^(1/2)-20*I)*(10+(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)))*(alpha/beta-1/(1/8*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)+1/8*(5-5^(1/2))^(1/2)*2^(1/2)+1/4*I-1/4*I*5^(1/2)))*(alpha/beta-1/(1/8*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)+1/8*(5-5^(1/2))^(1/2)*2^(1/2)-1/4*I+1/4*I*5^(1/2)))*(alpha/beta-4/(-3*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)-5*(5-5^(1/2))^(1/2)*2^(1/2)+20*I)*(-10+(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)))*(alpha/beta+4/(3*(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)+5*(5-5^(1/2))^(1/2)*2^(1/2)+20*I)*(-10+(5-5^(1/2))^(1/2)*5^(1/2)*2^(1/2)))*(alpha/beta-2/(10-2*5^(1/2))^(1/2)/(5+5^(1/2))*(10+5^(1/2)*(10-2*5^(1/2))^(1/2)))*(alpha/beta+2/(-5^(1/2)*(10-2*5^(1/2))^(1/2)+5*I+5*I*5^(1/2))*(10+5^(1/2)*(10-2*5^(1/2))^(1/2)))*(alpha/beta-1/(1/4*(10-2*5^(1/2))^(1/2)-1/4*I-1/4*I*5^(1/2)))*(alpha/beta-2/(5^(1/2)*(10-2*5^(1/2))^(1/2)+5*I+5*I*5^(1/2))*(10+5^(1/2)*(10-2*5^(1/2))^(1/2)))*(alpha/beta-1/(1/4*(10-2*5^(1/2))^(1/2)+1/4*I+1/4*I*5^(1/2)))*(alpha/beta-2/(10-2*5^(1/2))^(1/2)/(5+5^(1/2))*(-10+5^(1/2)*(10-2*5^(1/2))^(1/2)))*(alpha/beta-1/(-1/4*(10-2*5^(1/2))^(1/2)-1/4*I-1/4*I*5^(1/2)))*(alpha/beta-2/(5^(1/2)*(10-2*5^(1/2))^(1/2)+5*I+5*I*5^(1/2))*(-10+5^(1/2)*(10-2*5^(1/2))^(1/2)))*(alpha/beta-1/(-1/4*(10-2*5^(1/2))^(1/2)+1/4*I+1/4*I*5^(1/2)))*(alpha/beta+2/(-5^(1/2)*(10-2*5^(1/2))^(1/2)+5*I+5*I*5^(1/2))*(-10+5^(1/2)*(10-2*5^(1/2))^(1/2)))*(alpha/beta-I)*(alpha/beta+I)

Simplifying such expression we obtain 14411518807585587200000000000000 times the following polynomial

$$522\,{\alpha}^{25}{\beta}^{5}+{\beta}^{30}-10005\,{\beta}^{10}{\alpha}

^{20}-522\,{\beta}^{25}{\alpha}^{5}-10005\,{\beta}^{20}{\alpha}^{10}+{

\alpha}^{30}$$

which is precisely the second polynomial in the equation (27).