60 91 109 triangle

Right scalene triangle.

Sides: a = 60   b = 91   c = 109

Area: T = 2730
Perimeter: p = 260
Semiperimeter: s = 130

Angle ∠ A = α = 33.3988488468° = 33°23'55″ = 0.5832913589 rad
Angle ∠ B = β = 56.6021511532° = 56°36'5″ = 0.98878827378 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 91
Height: hb = 60
Height: hc = 50.09217431193

Median: ma = 95.81875349297
Median: mb = 75.30110624095
Median: mc = 54.5

Inradius: r = 21
Circumradius: R = 54.5

Vertex coordinates: A[109; 0] B[0; 0] C[33.02875229358; 50.09217431193]
Centroid: CG[47.34325076453; 16.69772477064]
Coordinates of the circumscribed circle: U[54.5; -0]
Coordinates of the inscribed circle: I[39; 21]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.6021511532° = 146°36'5″ = 0.5832913589 rad
∠ B' = β' = 123.3988488468° = 123°23'55″ = 0.98878827378 rad
∠ C' = γ' = 90° = 1.57107963268 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 = 60 ; ; b = 91 ; ; c = 109 ; ;

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

p = a+b+c = 60+91+109 = 260 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 260 }{ 2 } = 130 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 130 * (130-60)(130-91)(130-109) } ; ; T = sqrt{ 7452900 } = 2730 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2730 }{ 60 } = 91 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2730 }{ 91 } = 60 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2730 }{ 109 } = 50.09 ; ;

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{ 91**2+109**2-60**2 }{ 2 * 91 * 109 } ) = 33° 23'55" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 60**2+109**2-91**2 }{ 2 * 60 * 109 } ) = 56° 36'5" ; ; gamma = 180° - alpha - beta = 180° - 33° 23'55" - 56° 36'5" = 90° ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2730 }{ 130 } = 21 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 60 }{ 2 * sin 33° 23'55" } = 54.5 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 91**2+2 * 109**2 - 60**2 } }{ 2 } = 95.818 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 109**2+2 * 60**2 - 91**2 } }{ 2 } = 75.301 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 91**2+2 * 60**2 - 109**2 } }{ 2 } = 54.5 ; ;
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