19 23 25 triangle

Acute scalene triangle.

Sides: a = 19   b = 23   c = 25

Area: T = 208.2144282651
Perimeter: p = 67
Semiperimeter: s = 33.5

Angle ∠ A = α = 46.40442980945° = 46°24'15″ = 0.81099077888 rad
Angle ∠ B = β = 61.2465826369° = 61°14'45″ = 1.06989413232 rad
Angle ∠ C = γ = 72.35498755365° = 72°21' = 1.26327435415 rad

Height: ha = 21.91772929107
Height: hb = 18.10655897958
Height: hc = 16.65771426121

Median: ma = 22.06224114729
Median: mb = 18.99334199132
Median: mc = 16.9932645468

Inradius: r = 6.21553517209
Circumradius: R = 13.11774959048

Vertex coordinates: A[25; 0] B[0; 0] C[9.14; 16.65771426121]
Centroid: CG[11.38; 5.55223808707]
Coordinates of the circumscribed circle: U[12.5; 3.97772727858]
Coordinates of the inscribed circle: I[10.5; 6.21553517209]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.5965701906° = 133°35'45″ = 0.81099077888 rad
∠ B' = β' = 118.7544173631° = 118°45'15″ = 1.06989413232 rad
∠ C' = γ' = 107.6550124463° = 107°39' = 1.26327435415 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 = 23 ; ; c = 25 ; ;

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

p = a+b+c = 19+23+25 = 67 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 67 }{ 2 } = 33.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33.5 * (33.5-19)(33.5-23)(33.5-25) } ; ; T = sqrt{ 43353.19 } = 208.21 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 208.21 }{ 19 } = 21.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 208.21 }{ 23 } = 18.11 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 208.21 }{ 25 } = 16.66 ; ;

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**2-23**2-25**2 }{ 2 * 23 * 25 } ) = 46° 24'15" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-19**2-25**2 }{ 2 * 19 * 25 } ) = 61° 14'45" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 25**2-19**2-23**2 }{ 2 * 23 * 19 } ) = 72° 21' ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 208.21 }{ 33.5 } = 6.22 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 46° 24'15" } = 13.12 ; ;




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