17 19 23 triangle

Acute scalene triangle.

Sides: a = 17   b = 19   c = 23

Area: T = 158.6421695339
Perimeter: p = 59
Semiperimeter: s = 29.5

Angle ∠ A = α = 46.5566178098° = 46°33'22″ = 0.8132558595 rad
Angle ∠ B = β = 54.23994581681° = 54°14'22″ = 0.94766571295 rad
Angle ∠ C = γ = 79.20443637339° = 79°12'16″ = 1.38223769291 rad

Height: ha = 18.66437288634
Height: hb = 16.69991258251
Height: hc = 13.79549300295

Median: ma = 19.30767345763
Median: mb = 17.85435710714
Median: mc = 13.88334433769

Inradius: r = 5.37876845878
Circumradius: R = 11.70771996491

Vertex coordinates: A[23; 0] B[0; 0] C[9.93547826087; 13.79549300295]
Centroid: CG[10.97882608696; 4.59883100098]
Coordinates of the circumscribed circle: U[11.5; 2.19328346092]
Coordinates of the inscribed circle: I[10.5; 5.37876845878]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.4443821902° = 133°26'38″ = 0.8132558595 rad
∠ B' = β' = 125.7610541832° = 125°45'38″ = 0.94766571295 rad
∠ C' = γ' = 100.7965636266° = 100°47'44″ = 1.38223769291 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 = 19 ; ; c = 23 ; ;

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

p = a+b+c = 17+19+23 = 59 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 59 }{ 2 } = 29.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29.5 * (29.5-17)(29.5-19)(29.5-23) } ; ; T = sqrt{ 25167.19 } = 158.64 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 158.64 }{ 17 } = 18.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 158.64 }{ 19 } = 16.7 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 158.64 }{ 23 } = 13.79 ; ;

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**2-19**2-23**2 }{ 2 * 19 * 23 } ) = 46° 33'22" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-17**2-23**2 }{ 2 * 17 * 23 } ) = 54° 14'22" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-17**2-19**2 }{ 2 * 19 * 17 } ) = 79° 12'16" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 158.64 }{ 29.5 } = 5.38 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 17 }{ 2 * sin 46° 33'22" } = 11.71 ; ;




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