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How To Find the Area of a Triangle

Result

A = 1/4 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))

Definition

Defining inequalities

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)

Lamina properties

(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)

3

a>0 and b>0 and c>0 and a + b>c and b + c>a and a + c>b

(data not available)

sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))/(2 c)

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))

Mechanical properties

J_x invisible comma x = (-(a - b - c) (a + b - c) (a - b + c) (a + b + c))^(3/2)/(96 c^2)

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)

J_zz = -1/48 sqrt(-(a - b - c) (a + b - c) (a - b + c) (a + b + c)) (a^2 - 3 (b^2 + c^2))

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)

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))

Distance properties

a | b | c

p = a + b + c

r = 1/2 sqrt(-((a - b - c) (a + b - c) (a - b + c))/(a + b + c))

R = (a b c)/sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))

max(a, b, c)

χ = 1

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)

A^_ = 1/48 sqrt((a + b - c) (a - b + c) (-a + b + c) (a + b + c))

Alternate form

1/4 sqrt((-a - b - c) (a - b - c) (a + b - c) (a - b + c))

Alternate forms assuming a, b, and c are positive

1/4 sqrt(a + b - c) sqrt(a - b + c) sqrt(-a + b + c) sqrt(a + b + c) (-1)^(⌊-(arg(a + b - c) + arg(a - b + c) + arg(-a + b + c) - π)/(2 π)⌋)

1/4 sqrt(a + b - c) sqrt(a - b + c) sqrt(-a + b + c) sqrt(a + b + c) exp(i π floor(-arg(a + b - c)/(2 π) - arg(a - b + c)/(2 π) - arg(-a + b + c)/(2 π) + 1/2))

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