13 14 19 triangle

Acute scalene triangle.

Sides: a = 13   b = 14   c = 19

Area: T = 90.99545053286
Perimeter: p = 46
Semiperimeter: s = 23

Angle ∠ A = α = 43.17703054388° = 43°10'13″ = 0.7533463969 rad
Angle ∠ B = β = 47.45993311847° = 47°27'34″ = 0.828832159 rad
Angle ∠ C = γ = 89.37703633766° = 89°22'13″ = 1.56598070946 rad

Height: ha = 13.99991546659
Height: hb = 12.99992150469
Height: hc = 9.5788368982

Median: ma = 15.37704261489
Median: mb = 14.69769384567
Median: mc = 9.60546863561

Inradius: r = 3.95662828404
Circumradius: R = 9.50105736542

Vertex coordinates: A[19; 0] B[0; 0] C[8.78994736842; 9.5788368982]
Centroid: CG[9.26331578947; 3.19327896607]
Coordinates of the circumscribed circle: U[9.5; 0.10444019083]
Coordinates of the inscribed circle: I[9; 3.95662828404]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 136.8329694561° = 136°49'47″ = 0.7533463969 rad
∠ B' = β' = 132.5410668815° = 132°32'26″ = 0.828832159 rad
∠ C' = γ' = 90.63296366234° = 90°37'47″ = 1.56598070946 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 = 13 ; ; b = 14 ; ; c = 19 ; ;

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

p = a+b+c = 13+14+19 = 46 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 46 }{ 2 } = 23 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23 * (23-13)(23-14)(23-19) } ; ; T = sqrt{ 8280 } = 90.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 90.99 }{ 13 } = 14 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 90.99 }{ 14 } = 13 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 90.99 }{ 19 } = 9.58 ; ;

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{ 13**2-14**2-19**2 }{ 2 * 14 * 19 } ) = 43° 10'13" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 14**2-13**2-19**2 }{ 2 * 13 * 19 } ) = 47° 27'34" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 19**2-13**2-14**2 }{ 2 * 14 * 13 } ) = 89° 22'13" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 90.99 }{ 23 } = 3.96 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 43° 10'13" } = 9.5 ; ;




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