12 19 23 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 19   c = 23

Area: T = 113.8421995766
Perimeter: p = 54
Semiperimeter: s = 27

Angle ∠ A = α = 31.44004274006° = 31°24'2″ = 0.54880408447 rad
Angle ∠ B = β = 55.58326112896° = 55°34'57″ = 0.97700995739 rad
Angle ∠ C = γ = 93.01769613098° = 93°1'1″ = 1.62334522351 rad

Height: ha = 18.9743665961
Height: hb = 11.98333679754
Height: hc = 9.89993039797

Median: ma = 20.22437484162
Median: mb = 15.69223548265
Median: mc = 10.96658560997

Inradius: r = 4.21663702136
Circumradius: R = 11.51659611458

Vertex coordinates: A[23; 0] B[0; 0] C[6.78326086957; 9.89993039797]
Centroid: CG[9.92875362319; 3.32997679932]
Coordinates of the circumscribed circle: U[11.5; -0.60661032182]
Coordinates of the inscribed circle: I[8; 4.21663702136]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.6599572599° = 148°35'58″ = 0.54880408447 rad
∠ B' = β' = 124.417738871° = 124°25'3″ = 0.97700995739 rad
∠ C' = γ' = 86.98330386902° = 86°58'59″ = 1.62334522351 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 = 19 ; ; c = 23 ; ;

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

p = a+b+c = 12+19+23 = 54 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 54 }{ 2 } = 27 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27 * (27-12)(27-19)(27-23) } ; ; T = sqrt{ 12960 } = 113.84 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 113.84 }{ 12 } = 18.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 113.84 }{ 19 } = 11.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 113.84 }{ 23 } = 9.9 ; ;

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**2-19**2-23**2 }{ 2 * 19 * 23 } ) = 31° 24'2" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-12**2-23**2 }{ 2 * 12 * 23 } ) = 55° 34'57" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-12**2-19**2 }{ 2 * 19 * 12 } ) = 93° 1'1" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 113.84 }{ 27 } = 4.22 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 31° 24'2" } = 11.52 ; ;




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