13 17 19 triangle

Acute scalene triangle.

Sides: a = 13   b = 17   c = 19

Area: T = 107.8066249819
Perimeter: p = 49
Semiperimeter: s = 24.5

Angle ∠ A = α = 41.87767878296° = 41°52'36″ = 0.73108878278 rad
Angle ∠ B = β = 60.88003549713° = 60°48'1″ = 1.06111663806 rad
Angle ∠ C = γ = 77.32328571991° = 77°19'22″ = 1.35495384452 rad

Height: ha = 16.58655768952
Height: hb = 12.6833088214
Height: hc = 11.34880262967

Median: ma = 16.81551717208
Median: mb = 13.88334433769
Median: mc = 11.77992189894

Inradius: r = 4.44002550946
Circumradius: R = 9.73773760961

Vertex coordinates: A[19; 0] B[0; 0] C[6.34221052632; 11.34880262967]
Centroid: CG[8.44773684211; 3.78326754322]
Coordinates of the circumscribed circle: U[9.5; 2.13769354781]
Coordinates of the inscribed circle: I[7.5; 4.44002550946]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.123321217° = 138°7'24″ = 0.73108878278 rad
∠ B' = β' = 119.2199645029° = 119°11'59″ = 1.06111663806 rad
∠ C' = γ' = 102.6777142801° = 102°40'38″ = 1.35495384452 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 = 17 ; ; c = 19 ; ;

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

p = a+b+c = 13+17+19 = 49 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 49 }{ 2 } = 24.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24.5 * (24.5-13)(24.5-17)(24.5-19) } ; ; T = sqrt{ 11622.19 } = 107.81 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 107.81 }{ 13 } = 16.59 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 107.81 }{ 17 } = 12.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 107.81 }{ 19 } = 11.35 ; ;

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**2-17**2-19**2 }{ 2 * 17 * 19 } ) = 41° 52'36" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 17**2-13**2-19**2 }{ 2 * 13 * 19 } ) = 60° 48'1" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 19**2-13**2-17**2 }{ 2 * 17 * 13 } ) = 77° 19'22" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 107.81 }{ 24.5 } = 4.4 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 41° 52'36" } = 9.74 ; ;




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