9 21 23 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 21   c = 23

Area: T = 94.48437949069
Perimeter: p = 53
Semiperimeter: s = 26.5

Angle ∠ A = α = 23.03215063879° = 23°1'53″ = 0.40219756182 rad
Angle ∠ B = β = 65.90774000045° = 65°54'27″ = 1.15503011315 rad
Angle ∠ C = γ = 91.06110936076° = 91°3'40″ = 1.58993159039 rad

Height: ha = 20.99663988682
Height: hb = 8.99884566578
Height: hc = 8.21659821658

Median: ma = 21.55880611373
Median: mb = 13.9555285737
Median: mc = 11.34768057179

Inradius: r = 3.56554262229
Circumradius: R = 11.50219723866

Vertex coordinates: A[23; 0] B[0; 0] C[3.67439130435; 8.21659821658]
Centroid: CG[8.89113043478; 2.73986607219]
Coordinates of the circumscribed circle: U[11.5; -0.21329994886]
Coordinates of the inscribed circle: I[5.5; 3.56554262229]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 156.9688493612° = 156°58'7″ = 0.40219756182 rad
∠ B' = β' = 114.0932599996° = 114°5'33″ = 1.15503011315 rad
∠ C' = γ' = 88.93989063924° = 88°56'20″ = 1.58993159039 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 = 9 ; ; b = 21 ; ; c = 23 ; ;

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

p = a+b+c = 9+21+23 = 53 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 53 }{ 2 } = 26.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26.5 * (26.5-9)(26.5-21)(26.5-23) } ; ; T = sqrt{ 8927.19 } = 94.48 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 94.48 }{ 9 } = 21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 94.48 }{ 21 } = 9 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 94.48 }{ 23 } = 8.22 ; ;

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{ 9**2-21**2-23**2 }{ 2 * 21 * 23 } ) = 23° 1'53" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-9**2-23**2 }{ 2 * 9 * 23 } ) = 65° 54'27" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-9**2-21**2 }{ 2 * 21 * 9 } ) = 91° 3'40" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 94.48 }{ 26.5 } = 3.57 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 23° 1'53" } = 11.5 ; ;




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