14 15 19 triangle

Acute scalene triangle.

Sides: a = 14   b = 15   c = 19

Area: T = 103.9233048454
Perimeter: p = 48
Semiperimeter: s = 24

Angle ∠ A = α = 46.82664488927° = 46°49'35″ = 0.81772757102 rad
Angle ∠ B = β = 51.3876761809° = 51°23'12″ = 0.89768681855 rad
Angle ∠ C = γ = 81.78767892983° = 81°47'12″ = 1.42774487579 rad

Height: ha = 14.84661497792
Height: hb = 13.85664064606
Height: hc = 10.93992682583

Median: ma = 15.62204993518
Median: mb = 14.90880515159
Median: mc = 10.96658560997

Inradius: r = 4.33301270189
Circumradius: R = 9.59884482253

Vertex coordinates: A[19; 0] B[0; 0] C[8.73768421053; 10.93992682583]
Centroid: CG[9.24656140351; 3.64664227528]
Coordinates of the circumscribed circle: U[9.5; 1.37112068893]
Coordinates of the inscribed circle: I[9; 4.33301270189]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.1743551107° = 133°10'25″ = 0.81772757102 rad
∠ B' = β' = 128.6133238191° = 128°36'48″ = 0.89768681855 rad
∠ C' = γ' = 98.21332107017° = 98°12'48″ = 1.42774487579 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 = 15 ; ; c = 19 ; ;

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

p = a+b+c = 14+15+19 = 48 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 48 }{ 2 } = 24 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24 * (24-14)(24-15)(24-19) } ; ; T = sqrt{ 10800 } = 103.92 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 103.92 }{ 14 } = 14.85 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 103.92 }{ 15 } = 13.86 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 103.92 }{ 19 } = 10.94 ; ;

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**2-15**2-19**2 }{ 2 * 15 * 19 } ) = 46° 49'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15**2-14**2-19**2 }{ 2 * 14 * 19 } ) = 51° 23'12" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 19**2-14**2-15**2 }{ 2 * 15 * 14 } ) = 81° 47'12" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 103.92 }{ 24 } = 4.33 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 14 }{ 2 * sin 46° 49'35" } = 9.6 ; ;




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