65 72 96 triangle

Acute scalene triangle.

Sides: a = 65   b = 72   c = 96

Area: T = 2339.502249786
Perimeter: p = 233
Semiperimeter: s = 116.5

Angle ∠ A = α = 42.60549358939° = 42°36'18″ = 0.74435964089 rad
Angle ∠ B = β = 48.57765672248° = 48°34'36″ = 0.84878210374 rad
Angle ∠ C = γ = 88.81884968813° = 88°49'7″ = 1.55501752073 rad

Height: ha = 71.98546922419
Height: hb = 64.98661804962
Height: hc = 48.74396353721

Median: ma = 78.3822077033
Median: mb = 73.65112050139
Median: mc = 48.99548976935

Inradius: r = 20.08215665052
Circumradius: R = 48.01102073422

Vertex coordinates: A[96; 0] B[0; 0] C[43.00552083333; 48.74396353721]
Centroid: CG[46.33550694444; 16.2476545124]
Coordinates of the circumscribed circle: U[48; 0.99899540617]
Coordinates of the inscribed circle: I[44.5; 20.08215665052]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.3955064106° = 137°23'42″ = 0.74435964089 rad
∠ B' = β' = 131.4233432775° = 131°25'24″ = 0.84878210374 rad
∠ C' = γ' = 91.18215031187° = 91°10'53″ = 1.55501752073 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 = 65 ; ; b = 72 ; ; c = 96 ; ;

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

p = a+b+c = 65+72+96 = 233 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 233 }{ 2 } = 116.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 116.5 * (116.5-65)(116.5-72)(116.5-96) } ; ; T = sqrt{ 5473271.94 } = 2339.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2339.5 }{ 65 } = 71.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2339.5 }{ 72 } = 64.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2339.5 }{ 96 } = 48.74 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 72**2+96**2-65**2 }{ 2 * 72 * 96 } ) = 42° 36'18" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 65**2+96**2-72**2 }{ 2 * 65 * 96 } ) = 48° 34'36" ; ; gamma = 180° - alpha - beta = 180° - 42° 36'18" - 48° 34'36" = 88° 49'7" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2339.5 }{ 116.5 } = 20.08 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 65 }{ 2 * sin 42° 36'18" } = 48.01 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 72**2+2 * 96**2 - 65**2 } }{ 2 } = 78.382 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 96**2+2 * 65**2 - 72**2 } }{ 2 } = 73.651 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 72**2+2 * 65**2 - 96**2 } }{ 2 } = 48.995 ; ;
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