19 23 27 triangle

Acute scalene triangle.

Sides: a = 19   b = 23   c = 27

Area: T = 214.7610768065
Perimeter: p = 69
Semiperimeter: s = 34.5

Angle ∠ A = α = 43.76217426927° = 43°45'42″ = 0.76437864964 rad
Angle ∠ B = β = 56.8533365158° = 56°51'12″ = 0.99222784128 rad
Angle ∠ C = γ = 79.38548921493° = 79°23'6″ = 1.38655277443 rad

Height: ha = 22.60663966385
Height: hb = 18.6754849397
Height: hc = 15.90882050419

Median: ma = 23.21109887769
Median: mb = 20.31662496539
Median: mc = 16.21095650774

Inradius: r = 6.2254949799
Circumradius: R = 13.73550505242

Vertex coordinates: A[27; 0] B[0; 0] C[10.38988888889; 15.90882050419]
Centroid: CG[12.4632962963; 5.3032735014]
Coordinates of the circumscribed circle: U[13.5; 2.5330140886]
Coordinates of the inscribed circle: I[11.5; 6.2254949799]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 136.2388257307° = 136°14'18″ = 0.76437864964 rad
∠ B' = β' = 123.1476634842° = 123°8'48″ = 0.99222784128 rad
∠ C' = γ' = 100.6155107851° = 100°36'54″ = 1.38655277443 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 = 27 ; ;

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

p = a+b+c = 19+23+27 = 69 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 69 }{ 2 } = 34.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 34.5 * (34.5-19)(34.5-23)(34.5-27) } ; ; T = sqrt{ 46122.19 } = 214.76 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 214.76 }{ 19 } = 22.61 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 214.76 }{ 23 } = 18.67 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 214.76 }{ 27 } = 15.91 ; ;

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**2-27**2 }{ 2 * 23 * 27 } ) = 43° 45'42" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-19**2-27**2 }{ 2 * 19 * 27 } ) = 56° 51'12" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-19**2-23**2 }{ 2 * 23 * 19 } ) = 79° 23'6" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 214.76 }{ 34.5 } = 6.22 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 43° 45'42" } = 13.74 ; ;




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