14 19 23 triangle

Acute scalene triangle.

Sides: a = 14   b = 19   c = 23

Area: T = 132.8165661727
Perimeter: p = 56
Semiperimeter: s = 28

Angle ∠ A = α = 37.43443500203° = 37°26'4″ = 0.65333526612 rad
Angle ∠ B = β = 55.58326112896° = 55°34'57″ = 0.97700995739 rad
Angle ∠ C = γ = 86.98330386902° = 86°58'59″ = 1.51881404185 rad

Height: ha = 18.9743665961
Height: hb = 13.98105959713
Height: hc = 11.54991879763

Median: ma = 19.98997487421
Median: mb = 16.5
Median: mc = 12.09333866224

Inradius: r = 4.74334164903
Circumradius: R = 11.51659611458

Vertex coordinates: A[23; 0] B[0; 0] C[7.91330434783; 11.54991879763]
Centroid: CG[10.30443478261; 3.85497293254]
Coordinates of the circumscribed circle: U[11.5; 0.60661032182]
Coordinates of the inscribed circle: I[9; 4.74334164903]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 142.566564998° = 142°33'56″ = 0.65333526612 rad
∠ B' = β' = 124.417738871° = 124°25'3″ = 0.97700995739 rad
∠ C' = γ' = 93.01769613098° = 93°1'1″ = 1.51881404185 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 = 14 ; ; b = 19 ; ; c = 23 ; ;

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

p = a+b+c = 14+19+23 = 56 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 56 }{ 2 } = 28 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28 * (28-14)(28-19)(28-23) } ; ; T = sqrt{ 17640 } = 132.82 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 132.82 }{ 14 } = 18.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 132.82 }{ 19 } = 13.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 132.82 }{ 23 } = 11.55 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 14**2-19**2-23**2 }{ 2 * 19 * 23 } ) = 37° 26'4" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-14**2-23**2 }{ 2 * 14 * 23 } ) = 55° 34'57" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-14**2-19**2 }{ 2 * 19 * 14 } ) = 86° 58'59" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 132.82 }{ 28 } = 4.74 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 14 }{ 2 * sin 37° 26'4" } = 11.52 ; ;




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