19 19 21 triangle

Acute isosceles triangle.

Sides: a = 19   b = 19   c = 21

Area: T = 166.2688420032
Perimeter: p = 59
Semiperimeter: s = 29.5

Angle ∠ A = α = 56.45222617318° = 56°27'8″ = 0.98552778374 rad
Angle ∠ B = β = 56.45222617318° = 56°27'8″ = 0.98552778374 rad
Angle ∠ C = γ = 67.09554765365° = 67°5'44″ = 1.17110369788 rad

Height: ha = 17.50219389508
Height: hb = 17.50219389508
Height: hc = 15.83550876221

Median: ma = 17.62881025638
Median: mb = 17.62881025638
Median: mc = 15.83550876221

Inradius: r = 5.63662176282
Circumradius: R = 11.3998737052

Vertex coordinates: A[21; 0] B[0; 0] C[10.5; 15.83550876221]
Centroid: CG[10.5; 5.27883625407]
Coordinates of the circumscribed circle: U[10.5; 4.43663505701]
Coordinates of the inscribed circle: I[10.5; 5.63662176282]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 123.5487738268° = 123°32'52″ = 0.98552778374 rad
∠ B' = β' = 123.5487738268° = 123°32'52″ = 0.98552778374 rad
∠ C' = γ' = 112.9054523464° = 112°54'16″ = 1.17110369788 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 = 19 ; ; c = 21 ; ;

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

p = a+b+c = 19+19+21 = 59 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 59 }{ 2 } = 29.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29.5 * (29.5-19)(29.5-19)(29.5-21) } ; ; T = sqrt{ 27645.19 } = 166.27 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 166.27 }{ 19 } = 17.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 166.27 }{ 19 } = 17.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 166.27 }{ 21 } = 15.84 ; ;

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**2-19**2-21**2 }{ 2 * 19 * 21 } ) = 56° 27'8" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-19**2-21**2 }{ 2 * 19 * 21 } ) = 56° 27'8" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-19**2-19**2 }{ 2 * 19 * 19 } ) = 67° 5'44" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 166.27 }{ 29.5 } = 5.64 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 56° 27'8" } = 11.4 ; ;




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