19 20 23 triangle

Acute scalene triangle.

Sides: a = 19   b = 20   c = 23

Area: T = 180.9310926046
Perimeter: p = 62
Semiperimeter: s = 31

Angle ∠ A = α = 51.87441170479° = 51°52'27″ = 0.90553741391 rad
Angle ∠ B = β = 55.98998797635° = 55°54' = 0.97656369533 rad
Angle ∠ C = γ = 72.22660031886° = 72°13'34″ = 1.26105815612 rad

Height: ha = 19.04553606365
Height: hb = 18.09330926046
Height: hc = 15.7333124004

Median: ma = 19.34655421222
Median: mb = 18.5744175621
Median: mc = 15.75659512566

Inradius: r = 5.83664814854
Circumradius: R = 12.07664318613

Vertex coordinates: A[23; 0] B[0; 0] C[10.6522173913; 15.7333124004]
Centroid: CG[11.21773913043; 5.2444374668]
Coordinates of the circumscribed circle: U[11.5; 3.68664897261]
Coordinates of the inscribed circle: I[11; 5.83664814854]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 128.1265882952° = 128°7'33″ = 0.90553741391 rad
∠ B' = β' = 124.1100120236° = 124°6' = 0.97656369533 rad
∠ C' = γ' = 107.7743996811° = 107°46'26″ = 1.26105815612 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 = 19 ; ; b = 20 ; ; c = 23 ; ;

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

p = a+b+c = 19+20+23 = 62 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 62 }{ 2 } = 31 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31 * (31-19)(31-20)(31-23) } ; ; T = sqrt{ 32736 } = 180.93 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 180.93 }{ 19 } = 19.05 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 180.93 }{ 20 } = 18.09 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 180.93 }{ 23 } = 15.73 ; ;

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{ 19**2-20**2-23**2 }{ 2 * 20 * 23 } ) = 51° 52'27" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-19**2-23**2 }{ 2 * 19 * 23 } ) = 55° 54' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-19**2-20**2 }{ 2 * 20 * 19 } ) = 72° 13'34" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 180.93 }{ 31 } = 5.84 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 51° 52'27" } = 12.08 ; ;




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