19 22 23 triangle

Acute scalene triangle.

Sides: a = 19   b = 22   c = 23

Area: T = 193.4944185959
Perimeter: p = 64
Semiperimeter: s = 32

Angle ∠ A = α = 49.88991271742° = 49°53'21″ = 0.87107295301 rad
Angle ∠ B = β = 62.32201335598° = 62°19'12″ = 1.08876915209 rad
Angle ∠ C = γ = 67.7910739266° = 67°47'27″ = 1.18331716026 rad

Height: ha = 20.36878090483
Height: hb = 17.59903805417
Height: hc = 16.82655813878

Median: ma = 20.40222057631
Median: mb = 18
Median: mc = 17.03767250374

Inradius: r = 6.04766933112
Circumradius: R = 12.42215618577

Vertex coordinates: A[23; 0] B[0; 0] C[8.82660869565; 16.82655813878]
Centroid: CG[10.60986956522; 5.60985271293]
Coordinates of the circumscribed circle: U[11.5; 4.69552315156]
Coordinates of the inscribed circle: I[10; 6.04766933112]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130.1110872826° = 130°6'39″ = 0.87107295301 rad
∠ B' = β' = 117.687986644° = 117°40'48″ = 1.08876915209 rad
∠ C' = γ' = 112.2099260734° = 112°12'33″ = 1.18331716026 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 = 22 ; ; c = 23 ; ;

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

p = a+b+c = 19+22+23 = 64 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 64 }{ 2 } = 32 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 32 * (32-19)(32-22)(32-23) } ; ; T = sqrt{ 37440 } = 193.49 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 193.49 }{ 19 } = 20.37 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 193.49 }{ 22 } = 17.59 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 193.49 }{ 23 } = 16.83 ; ;

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**2-22**2-23**2 }{ 2 * 22 * 23 } ) = 49° 53'21" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-19**2-23**2 }{ 2 * 19 * 23 } ) = 62° 19'12" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-19**2-22**2 }{ 2 * 22 * 19 } ) = 67° 47'27" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 193.49 }{ 32 } = 6.05 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 49° 53'21" } = 12.42 ; ;




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