19 23 28 triangle

Acute scalene triangle.

Sides: a = 19   b = 23   c = 28

Area: T = 216.8877067388
Perimeter: p = 70
Semiperimeter: s = 35

Angle ∠ A = α = 42.34326054522° = 42°20'33″ = 0.7399017879 rad
Angle ∠ B = β = 54.62334598481° = 54°37'24″ = 0.95333592232 rad
Angle ∠ C = γ = 83.03439346997° = 83°2'2″ = 1.44992155514 rad

Height: ha = 22.83302176197
Height: hb = 18.86597449902
Height: hc = 15.49219333848

Median: ma = 23.79660080686
Median: mb = 20.98221352584
Median: mc = 15.78797338381

Inradius: r = 6.19767733539
Circumradius: R = 14.10441143524

Vertex coordinates: A[28; 0] B[0; 0] C[11; 15.49219333848]
Centroid: CG[13; 5.16439777949]
Coordinates of the circumscribed circle: U[14; 1.71105676446]
Coordinates of the inscribed circle: I[12; 6.19767733539]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.6577394548° = 137°39'27″ = 0.7399017879 rad
∠ B' = β' = 125.3776540152° = 125°22'36″ = 0.95333592232 rad
∠ C' = γ' = 96.96660653003° = 96°57'58″ = 1.44992155514 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 = 28 ; ;

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

p = a+b+c = 19+23+28 = 70 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 70 }{ 2 } = 35 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 35 * (35-19)(35-23)(35-28) } ; ; T = sqrt{ 47040 } = 216.89 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 216.89 }{ 19 } = 22.83 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 216.89 }{ 23 } = 18.86 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 216.89 }{ 28 } = 15.49 ; ;

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**2-28**2 }{ 2 * 23 * 28 } ) = 42° 20'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-19**2-28**2 }{ 2 * 19 * 28 } ) = 54° 37'24" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-19**2-23**2 }{ 2 * 23 * 19 } ) = 83° 2'2" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 216.89 }{ 35 } = 6.2 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 42° 20'33" } = 14.1 ; ;




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