16 17 23 triangle

Acute scalene triangle.

Sides: a = 16   b = 17   c = 23

Area: T = 135.9411163744
Perimeter: p = 56
Semiperimeter: s = 28

Angle ∠ A = α = 44.05552116096° = 44°3'19″ = 0.76989084953 rad
Angle ∠ B = β = 47.63302014306° = 47°37'49″ = 0.83113038384 rad
Angle ∠ C = γ = 88.31545869598° = 88°18'53″ = 1.541138032 rad

Height: ha = 16.9932645468
Height: hb = 15.99330780875
Height: hc = 11.82109707603

Median: ma = 18.5744175621
Median: mb = 17.89655301682
Median: mc = 11.84327192823

Inradius: r = 4.85550415623
Circumradius: R = 11.50549772779

Vertex coordinates: A[23; 0] B[0; 0] C[10.78326086957; 11.82109707603]
Centroid: CG[11.26108695652; 3.94403235868]
Coordinates of the circumscribed circle: U[11.5; 0.33883816846]
Coordinates of the inscribed circle: I[11; 4.85550415623]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.945478839° = 135°56'41″ = 0.76989084953 rad
∠ B' = β' = 132.3769798569° = 132°22'11″ = 0.83113038384 rad
∠ C' = γ' = 91.68554130402° = 91°41'7″ = 1.541138032 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 = 16 ; ; b = 17 ; ; c = 23 ; ;

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

p = a+b+c = 16+17+23 = 56 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 56 }{ 2 } = 28 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28 * (28-16)(28-17)(28-23) } ; ; T = sqrt{ 18480 } = 135.94 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 135.94 }{ 16 } = 16.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 135.94 }{ 17 } = 15.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 135.94 }{ 23 } = 11.82 ; ;

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{ 16**2-17**2-23**2 }{ 2 * 17 * 23 } ) = 44° 3'19" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 17**2-16**2-23**2 }{ 2 * 16 * 23 } ) = 47° 37'49" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-16**2-17**2 }{ 2 * 17 * 16 } ) = 88° 18'53" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 135.94 }{ 28 } = 4.86 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 16 }{ 2 * sin 44° 3'19" } = 11.5 ; ;




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