30-60-90 triangle | 3, 4, 5 triangle | Kepler triangle | scalene triangle
y>=0 and sqrt(3) a>=2 (sqrt(3) x + y) and x>=0
y>=0 and 12 a>=3 x + 4 y and x>=0
y>=0 and a>=x/sqrt(ϕ) + y and x>=0
y>=0 and y (a^2 + c^2) + x sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))<=c sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c)) + b^2 y and a^2 y + x sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))>=y (b^2 + c^2)
30-60-90 triangle | (0, 0) | (a/2, 0) | (0, (sqrt(3) a)/2) 3, 4, 5 triangle | (0, 3 a) | (0, 0) | (4 a, 0) Kepler triangle | (0, 0) | (sqrt(ϕ) a, 0) | (0, a) scalene triangle | (c, 0) | ((-a^2 + b^2 + c^2)/(2 c), sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(2 c)) | (0, 0)
30-60-90 triangle | 3 3, 4, 5 triangle | 3 Kepler triangle | 3 scalene triangle | 3
30-60-90 triangle | a>0 3, 4, 5 triangle | a>0 Kepler triangle | a>0 scalene triangle | a>0 and b>0 and c>0 and a + b>c and b + c>a and a + c>b and a!=b!=c
30-60-90 triangle | (sqrt(3) a)/2 3, 4, 5 triangle | 3 a Kepler triangle | a scalene triangle | sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(2 c)
30-60-90 triangle | A = (sqrt(3) a^2)/8 3, 4, 5 triangle | A = 6 a^2 Kepler triangle | A = (a^2 sqrt(ϕ))/2 scalene triangle | A = 1/4 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))
30-60-90 triangle | x^_ = (a/6, a/(2 sqrt(3))) 3, 4, 5 triangle | x^_ = ((4 a)/3, a) Kepler triangle | x^_ = ((sqrt(ϕ) a)/3, a/3) scalene triangle | x^_ = ((-a^2 + b^2 + 3 c^2)/(6 c), sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(6 c))
30-60-90 triangle | J_x invisible comma x = (sqrt(3) a^4)/64 3, 4, 5 triangle | J_x invisible comma x = 9 a^4 Kepler triangle | J_x invisible comma x = (a^4 sqrt(ϕ))/12 scalene triangle | J_x invisible comma x = ((a + b - c) (a - b + c) (-a + b + c) (a + b + c))^(3/2)/(96 c^2)
30-60-90 triangle | J_y invisible comma y = a^4/(64 sqrt(3)) 3, 4, 5 triangle | J_y invisible comma y = 16 a^4 Kepler triangle | J_y invisible comma y = 1/12 a^4 ϕ^(3/2) scalene triangle | J_y invisible comma y = (sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c)) (4 c^2 (b^2 - a^2) + (a^2 - b^2)^2 + 7 c^4))/(96 c^2)
30-60-90 triangle | J_zz = a^4/(16 sqrt(3)) 3, 4, 5 triangle | J_zz = 25 a^4 Kepler triangle | J_zz = 1/12 a^4 ϕ^(3/2) + (a^4 sqrt(ϕ))/12 scalene triangle | J_zz = -1/48 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c)) (a^2 - 3 (b^2 + c^2))
30-60-90 triangle | J_x invisible comma y = -a^4/128 3, 4, 5 triangle | J_x invisible comma y = -6 a^4 Kepler triangle | J_x invisible comma y = -(a^4 ϕ)/24 scalene triangle | J_x invisible comma y = -((a - b - c) (a + b - c) (a - b + c) (a + b + c) (a^2 - b^2 - 2 c^2))/(96 c^2)
30-60-90 triangle | r_x = a/(2 sqrt(2)) r_y = a/(2 sqrt(6)) 3, 4, 5 triangle | r_x = sqrt(3/2) a r_y = 2 sqrt(2/3) a Kepler triangle | r_x = a/sqrt(6) r_y = a sqrt(ϕ/6) scalene triangle | r_x = ((a + b - c) (a - b + c) (-a + b + c) (a + b + c))^(1/4)/(sqrt(6) c) r_y = sqrt(4 c^2 (b^2 - a^2) + (a^2 - b^2)^2 + 7 c^4)/(sqrt(6) c ((a + b - c) (a - b + c) (-a + b + c) (a + b + c))^(1/4))
30-60-90 triangle | a/2 | a | (sqrt(3) a)/2 3, 4, 5 triangle | 3 a | 4 a | 5 a Kepler triangle | a sqrt(ϕ) | a ϕ | a scalene triangle | a | b | c
30-60-90 triangle | a 3, 4, 5 triangle | 5 a Kepler triangle | a ϕ
30-60-90 triangle | p = 1/2 (3 + sqrt(3)) a 3, 4, 5 triangle | p = 12 a Kepler triangle | p = a (sqrt(ϕ) + ϕ + 1) scalene triangle | p = a + b + c
30-60-90 triangle | r = 1/4 (sqrt(3) - 1) a 3, 4, 5 triangle | r = a Kepler triangle | r = 1/2 a (ϕ - 1) (ϕ^(3/2) - 1) scalene triangle | r = 1/2 sqrt(-((a - b - c) (a + b - c) (a - b + c))/(a + b + c))
30-60-90 triangle | R = a/2 3, 4, 5 triangle | R = (5 a)/2 Kepler triangle | R = (a ϕ)/2 scalene triangle | R = (a b c)/sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))
30-60-90 triangle | a 3, 4, 5 triangle | 5 a Kepler triangle | a ϕ scalene triangle | max(a, b, c)
30-60-90 triangle | χ = 1 3, 4, 5 triangle | χ = 1 Kepler triangle | χ = 1 scalene triangle | χ = 1
30-60-90 triangle | s^_ = (a (204 + 36 sqrt(3) + 81 log(3) + 2 (9 + 8 sqrt(3)) log(2 + sqrt(3))))/1440 3, 4, 5 triangle | s^_ = (a (20460 + 9728 log(2) + 5103 log(3)))/22500 Kepler triangle | s^_ = (8 a (7 sqrt(2) + 3 sqrt(10) + 25 sqrt(1 + sqrt(5)) + 11 sqrt(5 (1 + sqrt(5))) + sqrt(2) (7 + 3 sqrt(5)) log(1/2 (1 + sqrt(5) + sqrt(2 (1 + sqrt(5))))) + sqrt(11 + 5 sqrt(5)) log(9 + 4 sqrt(5) + 2 sqrt(38 + 17 sqrt(5))) + sqrt(199 + 89 sqrt(5)) sinh^(-1)(2)))/(15 (1 + sqrt(5))^(9/2)) scalene triangle | s^_ = 2/15 (a + b + c) (1/2 (a + b + c) - a) (1/2 (a + b + c) - b) (1/2 (a + b + c) - c) (log((a + b + c)/(2 (1/2 (a + b + c) - a)))/a^3 + log((a + b + c)/(2 (1/2 (a + b + c) - b)))/b^3 + log((a + b + c)/(2 (1/2 (a + b + c) - c)))/c^3) + ((b - c)^2 (b + c))/(30 a^2) + ((c - a)^2 (a + c))/(30 b^2) + ((a + b) (a - b)^2)/(30 c^2) + 1/15 (a + b + c)
3, 4, 5 triangle | A^_ = a^2/2 Kepler triangle | A^_ = (a^2 sqrt(ϕ))/24 scalene triangle | A^_ = 1/48 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))
convex laminae | polygonal laminae | scalene triangular laminae | triangular laminae
