13 19 23 triangle

Acute scalene triangle.

Sides: a = 13   b = 19   c = 23

Area: T = 123.5499746963
Perimeter: p = 55
Semiperimeter: s = 27.5

Angle ∠ A = α = 34.4177308278° = 34°25'2″ = 0.60106953491 rad
Angle ∠ B = β = 55.69986751605° = 55°41'55″ = 0.97221252705 rad
Angle ∠ C = γ = 89.88440165614° = 89°53'2″ = 1.56987720339 rad

Height: ha = 198.9999610713
Height: hb = 132.9999733646
Height: hc = 10.73991084316

Median: ma = 20.06986322404
Median: mb = 16.08657079421
Median: mc = 11.52217186218

Inradius: r = 4.49108998896
Circumradius: R = 11.55000235622

Vertex coordinates: A[23; 0] B[0; 0] C[7.32660869565; 10.73991084316]
Centroid: CG[10.10986956522; 3.58797028105]
Coordinates of the circumscribed circle: U[11.5; 0.02332793999]
Coordinates of the inscribed circle: I[8.5; 4.49108998896]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 145.5832691722° = 145°34'58″ = 0.60106953491 rad
∠ B' = β' = 124.3011324839° = 124°18'5″ = 0.97221252705 rad
∠ C' = γ' = 90.11659834386° = 90°6'58″ = 1.56987720339 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 = 19 ; ; c = 23 ; ;

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

p = a+b+c = 13+19+23 = 55 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 55 }{ 2 } = 27.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27.5 * (27.5-13)(27.5-19)(27.5-23) } ; ; T = sqrt{ 15252.19 } = 123.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 123.5 }{ 13 } = 19 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 123.5 }{ 19 } = 13 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 123.5 }{ 23 } = 10.74 ; ;

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**2-19**2-23**2 }{ 2 * 19 * 23 } ) = 34° 25'2" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-13**2-23**2 }{ 2 * 13 * 23 } ) = 55° 41'55" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-13**2-19**2 }{ 2 * 19 * 13 } ) = 89° 53'2" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 123.5 }{ 27.5 } = 4.49 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 34° 25'2" } = 11.5 ; ;




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