17 20 23 triangle

Acute scalene triangle.

Sides: a = 17   b = 20   c = 23

Area: T = 165.2277116419
Perimeter: p = 60
Semiperimeter: s = 30

Angle ∠ A = α = 45.92107901521° = 45°55'15″ = 0.80114689833 rad
Angle ∠ B = β = 57.68881704787° = 57°41'17″ = 1.00768485143 rad
Angle ∠ C = γ = 76.39110393692° = 76°23'28″ = 1.3333275156 rad

Height: ha = 19.43884842845
Height: hb = 16.52327116419
Height: hc = 14.36875753407

Median: ma = 19.80553023203
Median: mb = 17.57883958312
Median: mc = 14.56988022843

Inradius: r = 5.50875705473
Circumradius: R = 11.83221982637

Vertex coordinates: A[23; 0] B[0; 0] C[9.08769565217; 14.36875753407]
Centroid: CG[10.69656521739; 4.78991917802]
Coordinates of the circumscribed circle: U[11.5; 2.78440466503]
Coordinates of the inscribed circle: I[10; 5.50875705473]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 134.0799209848° = 134°4'45″ = 0.80114689833 rad
∠ B' = β' = 122.3121829521° = 122°18'43″ = 1.00768485143 rad
∠ C' = γ' = 103.6098960631° = 103°36'32″ = 1.3333275156 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 = 17 ; ; b = 20 ; ; c = 23 ; ;

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

p = a+b+c = 17+20+23 = 60 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 60 }{ 2 } = 30 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30 * (30-17)(30-20)(30-23) } ; ; T = sqrt{ 27300 } = 165.23 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 165.23 }{ 17 } = 19.44 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 165.23 }{ 20 } = 16.52 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 165.23 }{ 23 } = 14.37 ; ;

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{ 17**2-20**2-23**2 }{ 2 * 20 * 23 } ) = 45° 55'15" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-17**2-23**2 }{ 2 * 17 * 23 } ) = 57° 41'17" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-17**2-20**2 }{ 2 * 20 * 17 } ) = 76° 23'28" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 165.23 }{ 30 } = 5.51 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 17 }{ 2 * sin 45° 55'15" } = 11.83 ; ;




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