19 23 24 triangle

Acute scalene triangle.

Sides: a = 19   b = 23   c = 24

Area: T = 203.9121745616
Perimeter: p = 66
Semiperimeter: s = 33

Angle ∠ A = α = 47.63302014306° = 47°37'49″ = 0.83113038384 rad
Angle ∠ B = β = 63.4255030481° = 63°25'30″ = 1.10769756101 rad
Angle ∠ C = γ = 68.94547680884° = 68°56'41″ = 1.20333132052 rad

Height: ha = 21.46443942753
Height: hb = 17.73114561405
Height: hc = 16.9932645468

Median: ma = 21.5
Median: mb = 18.33771208209
Median: mc = 17.34993515729

Inradius: r = 6.17991438065
Circumradius: R = 12.85985040165

Vertex coordinates: A[24; 0] B[0; 0] C[8.5; 16.9932645468]
Centroid: CG[10.83333333333; 5.6644215156]
Coordinates of the circumscribed circle: U[12; 4.62196456077]
Coordinates of the inscribed circle: I[10; 6.17991438065]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.3769798569° = 132°22'11″ = 0.83113038384 rad
∠ B' = β' = 116.5754969519° = 116°34'30″ = 1.10769756101 rad
∠ C' = γ' = 111.0555231912° = 111°3'19″ = 1.20333132052 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 = 19 ; ; b = 23 ; ; c = 24 ; ;

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

p = a+b+c = 19+23+24 = 66 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 66 }{ 2 } = 33 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33 * (33-19)(33-23)(33-24) } ; ; T = sqrt{ 41580 } = 203.91 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 203.91 }{ 19 } = 21.46 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 203.91 }{ 23 } = 17.73 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 203.91 }{ 24 } = 16.99 ; ;

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{ 19**2-23**2-24**2 }{ 2 * 23 * 24 } ) = 47° 37'49" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-19**2-24**2 }{ 2 * 19 * 24 } ) = 63° 25'30" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-19**2-23**2 }{ 2 * 23 * 19 } ) = 68° 56'41" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 203.91 }{ 33 } = 6.18 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 47° 37'49" } = 12.86 ; ;




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