15 19 23 triangle

Acute scalene triangle.

Sides: a = 15   b = 19   c = 23

Area: T = 141.7865709788
Perimeter: p = 57
Semiperimeter: s = 28.5

Angle ∠ A = α = 40.45990830808° = 40°27'33″ = 0.70661442121 rad
Angle ∠ B = β = 55.28800873965° = 55°16'48″ = 0.96548195359 rad
Angle ∠ C = γ = 84.26108295227° = 84°15'39″ = 1.47106289056 rad

Height: ha = 18.9054761305
Height: hb = 14.92548115566
Height: hc = 12.32991921555

Median: ma = 19.71767441531
Median: mb = 16.93436942219
Median: mc = 12.67987223331

Inradius: r = 4.97549371855
Circumradius: R = 11.55879348755

Vertex coordinates: A[23; 0] B[0; 0] C[8.54334782609; 12.32991921555]
Centroid: CG[10.51444927536; 4.11097307185]
Coordinates of the circumscribed circle: U[11.5; 1.15657934875]
Coordinates of the inscribed circle: I[9.5; 4.97549371855]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.5410916919° = 139°32'27″ = 0.70661442121 rad
∠ B' = β' = 124.7219912604° = 124°43'12″ = 0.96548195359 rad
∠ C' = γ' = 95.73991704773° = 95°44'21″ = 1.47106289056 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 = 15 ; ; b = 19 ; ; c = 23 ; ;

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

p = a+b+c = 15+19+23 = 57 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 57 }{ 2 } = 28.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28.5 * (28.5-15)(28.5-19)(28.5-23) } ; ; T = sqrt{ 20103.19 } = 141.79 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 141.79 }{ 15 } = 18.9 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 141.79 }{ 19 } = 14.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 141.79 }{ 23 } = 12.33 ; ;

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{ 15**2-19**2-23**2 }{ 2 * 19 * 23 } ) = 40° 27'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-15**2-23**2 }{ 2 * 15 * 23 } ) = 55° 16'48" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-15**2-19**2 }{ 2 * 19 * 15 } ) = 84° 15'39" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 141.79 }{ 28.5 } = 4.97 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 40° 27'33" } = 11.56 ; ;




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