90.5 106 135 triangle

Acute scalene triangle.

Sides: a = 90.5   b = 106   c = 135

Area: T = 4787.089937167
Perimeter: p = 331.5
Semiperimeter: s = 165.75

Angle ∠ A = α = 41.9944180209° = 41°59'39″ = 0.73329367113 rad
Angle ∠ B = β = 51.59554998995° = 51°35'44″ = 0.90105113525 rad
Angle ∠ C = γ = 86.41103198915° = 86°24'37″ = 1.50881445898 rad

Height: ha = 105.7922030313
Height: hb = 90.32224409749
Height: hc = 70.92198425432

Median: ma = 112.619854865
Median: mb = 101.9743648557
Median: mc = 71.81113848913

Inradius: r = 28.88113838411
Circumradius: R = 67.63326938695

Vertex coordinates: A[135; 0] B[0; 0] C[56.21994444444; 70.92198425432]
Centroid: CG[63.74398148148; 23.64399475144]
Coordinates of the circumscribed circle: U[67.5; 4.23545342182]
Coordinates of the inscribed circle: I[59.75; 28.88113838411]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.0065819791° = 138°21″ = 0.73329367113 rad
∠ B' = β' = 128.4054500101° = 128°24'16″ = 0.90105113525 rad
∠ C' = γ' = 93.59896801085° = 93°35'23″ = 1.50881445898 rad

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How did we calculate this triangle?

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

p = a+b+c = 90.5+106+135 = 331.5 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 331.5 }{ 2 } = 165.75 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 165.75 * (165.75-90.5)(165.75-106)(165.75-135) } ; ; T = sqrt{ 22916224.65 } = 4787.09 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4787.09 }{ 90.5 } = 105.79 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4787.09 }{ 106 } = 90.32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4787.09 }{ 135 } = 70.92 ; ;

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{ 106**2+135**2-90.5**2 }{ 2 * 106 * 135 } ) = 41° 59'39" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 90.5**2+135**2-106**2 }{ 2 * 90.5 * 135 } ) = 51° 35'44" ; ; gamma = 180° - alpha - beta = 180° - 41° 59'39" - 51° 35'44" = 86° 24'37" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4787.09 }{ 165.75 } = 28.88 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 90.5 }{ 2 * sin 41° 59'39" } = 67.63 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 106**2+2 * 135**2 - 90.5**2 } }{ 2 } = 112.619 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 135**2+2 * 90.5**2 - 106**2 } }{ 2 } = 101.974 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 106**2+2 * 90.5**2 - 135**2 } }{ 2 } = 71.811 ; ;
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