14 21 23 triangle

Acute scalene triangle.

Sides: a = 14   b = 21   c = 23

Area: T = 144.4999134946
Perimeter: p = 58
Semiperimeter: s = 29

Angle ∠ A = α = 36.75111085714° = 36°45'4″ = 0.64114278483 rad
Angle ∠ B = β = 63.83326940251° = 63°49'58″ = 1.11440906812 rad
Angle ∠ C = γ = 79.41661974035° = 79°24'58″ = 1.38660741241 rad

Height: ha = 20.64327335636
Height: hb = 13.76218223758
Height: hc = 12.56551421692

Median: ma = 20.88106130178
Median: mb = 15.88223801743
Median: mc = 13.6477344064

Inradius: r = 4.98327287912
Circumradius: R = 11.69990319744

Vertex coordinates: A[23; 0] B[0; 0] C[6.17439130435; 12.56551421692]
Centroid: CG[9.72546376812; 4.18883807231]
Coordinates of the circumscribed circle: U[11.5; 2.14988017912]
Coordinates of the inscribed circle: I[8; 4.98327287912]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 143.2498891429° = 143°14'56″ = 0.64114278483 rad
∠ B' = β' = 116.1677305975° = 116°10'2″ = 1.11440906812 rad
∠ C' = γ' = 100.5843802597° = 100°35'2″ = 1.38660741241 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 = 14 ; ; b = 21 ; ; c = 23 ; ;

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

p = a+b+c = 14+21+23 = 58 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 58 }{ 2 } = 29 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29 * (29-14)(29-21)(29-23) } ; ; T = sqrt{ 20880 } = 144.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 144.5 }{ 14 } = 20.64 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 144.5 }{ 21 } = 13.76 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 144.5 }{ 23 } = 12.57 ; ;

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{ 14**2-21**2-23**2 }{ 2 * 21 * 23 } ) = 36° 45'4" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-14**2-23**2 }{ 2 * 14 * 23 } ) = 63° 49'58" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-14**2-21**2 }{ 2 * 21 * 14 } ) = 79° 24'58" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 144.5 }{ 29 } = 4.98 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 14 }{ 2 * sin 36° 45'4" } = 11.7 ; ;




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