9 20 23 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 20   c = 23

Area: T = 89.19664124839
Perimeter: p = 52
Semiperimeter: s = 26

Angle ∠ A = α = 22.81883300648° = 22°49'6″ = 0.39882549894 rad
Angle ∠ B = β = 59.51994142744° = 59°31'10″ = 1.03988097479 rad
Angle ∠ C = γ = 97.66222556608° = 97°39'44″ = 1.70545279162 rad

Height: ha = 19.82114249964
Height: hb = 8.92196412484
Height: hc = 7.75662097812

Median: ma = 21.07772389084
Median: mb = 14.31878210633
Median: mc = 10.40443260233

Inradius: r = 3.43106312494
Circumradius: R = 11.60436056964

Vertex coordinates: A[23; 0] B[0; 0] C[4.56552173913; 7.75662097812]
Centroid: CG[9.18884057971; 2.58554032604]
Coordinates of the circumscribed circle: U[11.5; -1.54771474262]
Coordinates of the inscribed circle: I[6; 3.43106312494]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 157.1821669935° = 157°10'54″ = 0.39882549894 rad
∠ B' = β' = 120.4810585726° = 120°28'50″ = 1.03988097479 rad
∠ C' = γ' = 82.33877443392° = 82°20'16″ = 1.70545279162 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 = 20 ; ; c = 23 ; ;

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

p = a+b+c = 9+20+23 = 52 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 52 }{ 2 } = 26 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26 * (26-9)(26-20)(26-23) } ; ; T = sqrt{ 7956 } = 89.2 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 89.2 }{ 9 } = 19.82 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 89.2 }{ 20 } = 8.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 89.2 }{ 23 } = 7.76 ; ;

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**2-20**2-23**2 }{ 2 * 20 * 23 } ) = 22° 49'6" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-9**2-23**2 }{ 2 * 9 * 23 } ) = 59° 31'10" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-9**2-20**2 }{ 2 * 20 * 9 } ) = 97° 39'44" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 89.2 }{ 26 } = 3.43 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 22° 49'6" } = 11.6 ; ;




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