12 14 19 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 14   c = 19

Area: T = 83.8365776969
Perimeter: p = 45
Semiperimeter: s = 22.5

Angle ∠ A = α = 39.07655206502° = 39°4'32″ = 0.68219964923 rad
Angle ∠ B = β = 47.34111576513° = 47°20'28″ = 0.82662590727 rad
Angle ∠ C = γ = 93.58333216985° = 93°35' = 1.63333370886 rad

Height: ha = 13.97326294948
Height: hb = 11.9776539567
Height: hc = 8.82548186283

Median: ma = 15.57224115024
Median: mb = 14.26553426177
Median: mc = 8.93302855497

Inradius: r = 3.7266034532
Circumradius: R = 9.51986092245

Vertex coordinates: A[19; 0] B[0; 0] C[8.13215789474; 8.82548186283]
Centroid: CG[9.04438596491; 2.94216062094]
Coordinates of the circumscribed circle: U[9.5; -0.59549130765]
Coordinates of the inscribed circle: I[8.5; 3.7266034532]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140.924447935° = 140°55'28″ = 0.68219964923 rad
∠ B' = β' = 132.6598842349° = 132°39'32″ = 0.82662590727 rad
∠ C' = γ' = 86.41766783015° = 86°25' = 1.63333370886 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 = 12 ; ; b = 14 ; ; c = 19 ; ;

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

p = a+b+c = 12+14+19 = 45 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 45 }{ 2 } = 22.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22.5 * (22.5-12)(22.5-14)(22.5-19) } ; ; T = sqrt{ 7028.44 } = 83.84 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 83.84 }{ 12 } = 13.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 83.84 }{ 14 } = 11.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 83.84 }{ 19 } = 8.82 ; ;

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{ 12**2-14**2-19**2 }{ 2 * 14 * 19 } ) = 39° 4'32" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 14**2-12**2-19**2 }{ 2 * 12 * 19 } ) = 47° 20'28" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 19**2-12**2-14**2 }{ 2 * 14 * 12 } ) = 93° 35' ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 83.84 }{ 22.5 } = 3.73 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 39° 4'32" } = 9.52 ; ;




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