9 19 23 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 19   c = 23

Area: T = 82.68772874147
Perimeter: p = 51
Semiperimeter: s = 25.5

Angle ∠ A = α = 22.23765612709° = 22°14'12″ = 0.38881012085 rad
Angle ∠ B = β = 53.02662349852° = 53°1'34″ = 0.92554823904 rad
Angle ∠ C = γ = 104.7377203744° = 104°44'14″ = 1.82880090547 rad

Height: ha = 18.37549527588
Height: hb = 8.7043924991
Height: hc = 7.19901989056

Median: ma = 20.60994638455
Median: mb = 14.65443508898
Median: mc = 9.42107218407

Inradius: r = 3.24326387221
Circumradius: R = 11.89111870342

Vertex coordinates: A[23; 0] B[0; 0] C[5.41330434783; 7.19901989056]
Centroid: CG[9.47110144928; 2.39767329685]
Coordinates of the circumscribed circle: U[11.5; -3.02549510877]
Coordinates of the inscribed circle: I[6.5; 3.24326387221]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 157.7633438729° = 157°45'48″ = 0.38881012085 rad
∠ B' = β' = 126.9743765015° = 126°58'26″ = 0.92554823904 rad
∠ C' = γ' = 75.2632796256° = 75°15'46″ = 1.82880090547 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 = 9 ; ; b = 19 ; ; c = 23 ; ;

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

p = a+b+c = 9+19+23 = 51 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 51 }{ 2 } = 25.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25.5 * (25.5-9)(25.5-19)(25.5-23) } ; ; T = sqrt{ 6837.19 } = 82.69 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 82.69 }{ 9 } = 18.37 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 82.69 }{ 19 } = 8.7 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 82.69 }{ 23 } = 7.19 ; ;

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{ 9**2-19**2-23**2 }{ 2 * 19 * 23 } ) = 22° 14'12" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-9**2-23**2 }{ 2 * 9 * 23 } ) = 53° 1'34" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-9**2-19**2 }{ 2 * 19 * 9 } ) = 104° 44'14" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 82.69 }{ 25.5 } = 3.24 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 22° 14'12" } = 11.89 ; ;




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