18 19 23 triangle

Acute scalene triangle.

Sides: a = 18   b = 19   c = 23

Area: T = 166.4933243106
Perimeter: p = 60
Semiperimeter: s = 30

Angle ∠ A = α = 49.6399310869° = 49°38'22″ = 0.86663694131 rad
Angle ∠ B = β = 53.54441706172° = 53°32'39″ = 0.93545220725 rad
Angle ∠ C = γ = 76.81765185138° = 76°48'59″ = 1.3410701168 rad

Height: ha = 18.4999249234
Height: hb = 17.52656045375
Height: hc = 14.47876733136

Median: ma = 19.07987840283
Median: mb = 18.33771208209
Median: mc = 14.5

Inradius: r = 5.55497747702
Circumradius: R = 11.81112901359

Vertex coordinates: A[23; 0] B[0; 0] C[10.69656521739; 14.47876733136]
Centroid: CG[11.2321884058; 4.82658911045]
Coordinates of the circumscribed circle: U[11.5; 2.69438030135]
Coordinates of the inscribed circle: I[11; 5.55497747702]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130.3610689131° = 130°21'38″ = 0.86663694131 rad
∠ B' = β' = 126.4565829383° = 126°27'21″ = 0.93545220725 rad
∠ C' = γ' = 103.1833481486° = 103°11'1″ = 1.3410701168 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 = 18 ; ; b = 19 ; ; c = 23 ; ;

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

p = a+b+c = 18+19+23 = 60 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 60 }{ 2 } = 30 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30 * (30-18)(30-19)(30-23) } ; ; T = sqrt{ 27720 } = 166.49 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 166.49 }{ 18 } = 18.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 166.49 }{ 19 } = 17.53 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 166.49 }{ 23 } = 14.48 ; ;

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{ 18**2-19**2-23**2 }{ 2 * 19 * 23 } ) = 49° 38'22" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-18**2-23**2 }{ 2 * 18 * 23 } ) = 53° 32'39" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-18**2-19**2 }{ 2 * 19 * 18 } ) = 76° 48'59" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 166.49 }{ 30 } = 5.55 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 18 }{ 2 * sin 49° 38'22" } = 11.81 ; ;




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