12.81 19.21 23.09 triangle

Obtuse scalene triangle.

Sides: a = 12.81   b = 19.21   c = 23.09

Area: T = 123.0440049802
Perimeter: p = 55.11
Semiperimeter: s = 27.555

Angle ∠ A = α = 33.69659501308° = 33°41'45″ = 0.58881052744 rad
Angle ∠ B = β = 56.30108018369° = 56°18'3″ = 0.98326343636 rad
Angle ∠ C = γ = 90.00332480323° = 90°12″ = 1.57108530157 rad

Height: ha = 19.21099999691
Height: hb = 12.81099999794
Height: hc = 10.65774317715

Median: ma = 20.25499895062
Median: mb = 16.0111435757
Median: mc = 11.54443958265

Inradius: r = 4.46552531229
Circumradius: R = 11.54550000186

Vertex coordinates: A[23.09; 0] B[0; 0] C[7.10774079688; 10.65774317715]
Centroid: CG[10.06658026563; 3.55224772572]
Coordinates of the circumscribed circle: U[11.545; -0.00106544729]
Coordinates of the inscribed circle: I[8.345; 4.46552531229]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.3044049869° = 146°18'15″ = 0.58881052744 rad
∠ B' = β' = 123.6999198163° = 123°41'57″ = 0.98326343636 rad
∠ C' = γ' = 89.99767519677° = 89°59'48″ = 1.57108530157 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 = 12.81+19.21+23.09 = 55.11 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 55.11 }{ 2 } = 27.56 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27.56 * (27.56-12.81)(27.56-19.21)(27.56-23.09) } ; ; T = sqrt{ 15138.85 } = 123.04 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 123.04 }{ 12.81 } = 19.21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 123.04 }{ 19.21 } = 12.81 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 123.04 }{ 23.09 } = 10.66 ; ;

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{ 19.21**2+23.09**2-12.81**2 }{ 2 * 19.21 * 23.09 } ) = 33° 41'45" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12.81**2+23.09**2-19.21**2 }{ 2 * 12.81 * 23.09 } ) = 56° 18'3" ; ;
 gamma = 180° - alpha - beta = 180° - 33° 41'45" - 56° 18'3" = 90° 12" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 123.04 }{ 27.56 } = 4.47 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 12.81 }{ 2 * sin 33° 41'45" } = 11.55 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 19.21**2+2 * 23.09**2 - 12.81**2 } }{ 2 } = 20.25 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 23.09**2+2 * 12.81**2 - 19.21**2 } }{ 2 } = 16.011 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 19.21**2+2 * 12.81**2 - 23.09**2 } }{ 2 } = 11.544 ; ;
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