40 70 100 triangle

Obtuse scalene triangle.

Sides: a = 40   b = 70   c = 100

Area: T = 1092.875464972
Perimeter: p = 210
Semiperimeter: s = 105

Angle ∠ A = α = 18.19548723388° = 18°11'42″ = 0.31875604293 rad
Angle ∠ B = β = 33.12329402077° = 33°7'23″ = 0.57881043646 rad
Angle ∠ C = γ = 128.6822187453° = 128°40'56″ = 2.24659278597 rad

Height: ha = 54.6443732486
Height: hb = 31.2254989992
Height: hc = 21.85774929944

Median: ma = 83.96442781187
Median: mb = 67.63987462923
Median: mc = 27.38661278753

Inradius: r = 10.40883299973
Circumradius: R = 64.0511261522

Vertex coordinates: A[100; 0] B[0; 0] C[33.5; 21.85774929944]
Centroid: CG[44.5; 7.28658309981]
Coordinates of the circumscribed circle: U[50; -40.03220384513]
Coordinates of the inscribed circle: I[35; 10.40883299973]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 161.8055127661° = 161°48'18″ = 0.31875604293 rad
∠ B' = β' = 146.8777059792° = 146°52'37″ = 0.57881043646 rad
∠ C' = γ' = 51.31878125465° = 51°19'4″ = 2.24659278597 rad

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How did we calculate this triangle?

Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 40 ; ; b = 70 ; ; c = 100 ; ;

1. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 40+70+100 = 210 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 210 }{ 2 } = 105 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 105 * (105-40)(105-70)(105-100) } ; ; T = sqrt{ 1194375 } = 1092.87 ; ;

4. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1092.87 }{ 40 } = 54.64 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1092.87 }{ 70 } = 31.22 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1092.87 }{ 100 } = 21.86 ; ;

5. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 70**2+100**2-40**2 }{ 2 * 70 * 100 } ) = 18° 11'42" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 40**2+100**2-70**2 }{ 2 * 40 * 100 } ) = 33° 7'23" ; ; gamma = 180° - alpha - beta = 180° - 18° 11'42" - 33° 7'23" = 128° 40'56" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1092.87 }{ 105 } = 10.41 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 40 }{ 2 * sin 18° 11'42" } = 64.05 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 70**2+2 * 100**2 - 40**2 } }{ 2 } = 83.964 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 100**2+2 * 40**2 - 70**2 } }{ 2 } = 67.639 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 70**2+2 * 40**2 - 100**2 } }{ 2 } = 27.386 ; ;
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