13 17 23 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 17   c = 23

Area: T = 109.0655060858
Perimeter: p = 53
Semiperimeter: s = 26.5

Angle ∠ A = α = 33.909914284° = 33°54'33″ = 0.59218261891 rad
Angle ∠ B = β = 46.84771893515° = 46°50'50″ = 0.81876376995 rad
Angle ∠ C = γ = 99.24436678085° = 99°14'37″ = 1.7322128765 rad

Height: ha = 16.7799240132
Height: hb = 12.83111836304
Height: hc = 9.48439183355

Median: ma = 19.15107180022
Median: mb = 16.63658047596
Median: mc = 9.83661577865

Inradius: r = 4.11656626739
Circumradius: R = 11.65113023511

Vertex coordinates: A[23; 0] B[0; 0] C[8.89113043478; 9.48439183355]
Centroid: CG[10.63304347826; 3.16113061118]
Coordinates of the circumscribed circle: U[11.5; -1.87215892917]
Coordinates of the inscribed circle: I[9.5; 4.11656626739]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.091085716° = 146°5'27″ = 0.59218261891 rad
∠ B' = β' = 133.1532810649° = 133°9'10″ = 0.81876376995 rad
∠ C' = γ' = 80.75663321915° = 80°45'23″ = 1.7322128765 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 = 13 ; ; b = 17 ; ; c = 23 ; ;

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

p = a+b+c = 13+17+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 * (26.5-13)(26.5-17)(26.5-23) } ; ; T = sqrt{ 11895.19 } = 109.07 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 109.07 }{ 13 } = 16.78 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 109.07 }{ 17 } = 12.83 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 109.07 }{ 23 } = 9.48 ; ;

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{ 13**2-17**2-23**2 }{ 2 * 17 * 23 } ) = 33° 54'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 17**2-13**2-23**2 }{ 2 * 13 * 23 } ) = 46° 50'50" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-13**2-17**2 }{ 2 * 17 * 13 } ) = 99° 14'37" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 109.07 }{ 26.5 } = 4.12 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 33° 54'33" } = 11.65 ; ;




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