19 21 26 triangle

Acute scalene triangle.

Sides: a = 19   b = 21   c = 26

Area: T = 196.9977461913
Perimeter: p = 66
Semiperimeter: s = 33

Angle ∠ A = α = 46.18769385396° = 46°11'13″ = 0.80661141489 rad
Angle ∠ B = β = 52.89877817478° = 52°53'52″ = 0.92332404585 rad
Angle ∠ C = γ = 80.91552797126° = 80°54'55″ = 1.41222380462 rad

Height: ha = 20.73765749382
Height: hb = 18.76216630393
Height: hc = 15.15436509164

Median: ma = 21.63990850084
Median: mb = 20.20551973512
Median: mc = 15.23215462117

Inradius: r = 5.9769620058
Circumradius: R = 13.1655144235

Vertex coordinates: A[26; 0] B[0; 0] C[11.46215384615; 15.15436509164]
Centroid: CG[12.48771794872; 5.05112169721]
Coordinates of the circumscribed circle: U[13; 2.07987069845]
Coordinates of the inscribed circle: I[12; 5.9769620058]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.813306146° = 133°48'47″ = 0.80661141489 rad
∠ B' = β' = 127.1022218252° = 127°6'8″ = 0.92332404585 rad
∠ C' = γ' = 99.08547202874° = 99°5'5″ = 1.41222380462 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 = 26 ; ;

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

p = a+b+c = 19+21+26 = 66 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 66 }{ 2 } = 33 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33 * (33-19)(33-21)(33-26) } ; ; T = sqrt{ 38808 } = 197 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 197 }{ 19 } = 20.74 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 197 }{ 21 } = 18.76 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 197 }{ 26 } = 15.15 ; ;

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**2-26**2 }{ 2 * 21 * 26 } ) = 46° 11'13" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-19**2-26**2 }{ 2 * 19 * 26 } ) = 52° 53'52" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 26**2-19**2-21**2 }{ 2 * 21 * 19 } ) = 80° 54'55" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 197 }{ 33 } = 5.97 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 46° 11'13" } = 13.17 ; ;




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