19 21 23 triangle

Acute scalene triangle.

Sides: a = 19   b = 21   c = 23

Area: T = 187.4622496249
Perimeter: p = 63
Semiperimeter: s = 31.5

Angle ∠ A = α = 50.91877927026° = 50°55'4″ = 0.88986831305 rad
Angle ∠ B = β = 59.08773973855° = 59°5'15″ = 1.03112696308 rad
Angle ∠ C = γ = 69.99548099118° = 69°59'41″ = 1.22216398923 rad

Height: ha = 19.7332894342
Height: hb = 17.85435710714
Height: hc = 16.30110866304

Median: ma = 19.86883164863
Median: mb = 18.29661744635
Median: mc = 16.39435963108

Inradius: r = 5.95111903571
Circumradius: R = 12.23884479344

Vertex coordinates: A[23; 0] B[0; 0] C[9.76108695652; 16.30110866304]
Centroid: CG[10.92202898551; 5.43436955435]
Coordinates of the circumscribed circle: U[11.5; 4.18768374512]
Coordinates of the inscribed circle: I[10.5; 5.95111903571]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 129.0822207297° = 129°4'56″ = 0.88986831305 rad
∠ B' = β' = 120.9132602614° = 120°54'45″ = 1.03112696308 rad
∠ C' = γ' = 110.0055190088° = 110°19″ = 1.22216398923 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 = 21 ; ; c = 23 ; ;

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

p = a+b+c = 19+21+23 = 63 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 63 }{ 2 } = 31.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31.5 * (31.5-19)(31.5-21)(31.5-23) } ; ; T = sqrt{ 35142.19 } = 187.46 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 187.46 }{ 19 } = 19.73 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 187.46 }{ 21 } = 17.85 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 187.46 }{ 23 } = 16.3 ; ;

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**2-21**2-23**2 }{ 2 * 21 * 23 } ) = 50° 55'4" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-19**2-23**2 }{ 2 * 19 * 23 } ) = 59° 5'15" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-19**2-21**2 }{ 2 * 21 * 19 } ) = 69° 59'41" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 187.46 }{ 31.5 } = 5.95 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 50° 55'4" } = 12.24 ; ;




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