19 21 24 triangle

Acute scalene triangle.

Sides: a = 19   b = 21   c = 24

Area: T = 191.332217189
Perimeter: p = 64
Semiperimeter: s = 32

Angle ∠ A = α = 49.39985335° = 49°23'55″ = 0.86221670552 rad
Angle ∠ B = β = 57.0533229556° = 57°3'12″ = 0.99657667046 rad
Angle ∠ C = γ = 73.5488236944° = 73°32'54″ = 1.28436588937 rad

Height: ha = 20.144022862
Height: hb = 18.22221116085
Height: hc = 15.94443476575

Median: ma = 20.45111613362
Median: mb = 18.92774932307
Median: mc = 16.03112195419

Inradius: r = 5.97991303716
Circumradius: R = 12.51222710747

Vertex coordinates: A[24; 0] B[0; 0] C[10.33333333333; 15.94443476575]
Centroid: CG[11.44444444444; 5.31547825525]
Coordinates of the circumscribed circle: U[12; 3.54435755174]
Coordinates of the inscribed circle: I[11; 5.97991303716]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130.60114665° = 130°36'5″ = 0.86221670552 rad
∠ B' = β' = 122.9476770444° = 122°56'48″ = 0.99657667046 rad
∠ C' = γ' = 106.4521763056° = 106°27'6″ = 1.28436588937 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 = 24 ; ;

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

p = a+b+c = 19+21+24 = 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)(32-21)(32-24) } ; ; T = sqrt{ 36608 } = 191.33 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 191.33 }{ 19 } = 20.14 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 191.33 }{ 21 } = 18.22 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 191.33 }{ 24 } = 15.94 ; ;

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**2-24**2 }{ 2 * 21 * 24 } ) = 49° 23'55" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-19**2-24**2 }{ 2 * 19 * 24 } ) = 57° 3'12" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-19**2-21**2 }{ 2 * 21 * 19 } ) = 73° 32'54" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 191.33 }{ 32 } = 5.98 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 49° 23'55" } = 12.51 ; ;




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