19 19 23 triangle

Acute isosceles triangle.

Sides: a = 19   b = 19   c = 23

Area: T = 173.9321559816
Perimeter: p = 61
Semiperimeter: s = 30.5

Angle ∠ A = α = 52.75222191042° = 52°45'8″ = 0.92106999111 rad
Angle ∠ B = β = 52.75222191042° = 52°45'8″ = 0.92106999111 rad
Angle ∠ C = γ = 74.49655617917° = 74°29'44″ = 1.33001928314 rad

Height: ha = 18.30985852438
Height: hb = 18.30985852438
Height: hc = 15.12444834623

Median: ma = 18.83548082018
Median: mb = 18.83548082018
Median: mc = 15.12444834623

Inradius: r = 5.70326740923
Circumradius: R = 11.93442918686

Vertex coordinates: A[23; 0] B[0; 0] C[11.5; 15.12444834623]
Centroid: CG[11.5; 5.04114944874]
Coordinates of the circumscribed circle: U[11.5; 3.19901915937]
Coordinates of the inscribed circle: I[11.5; 5.70326740923]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 127.2487780896° = 127°14'52″ = 0.92106999111 rad
∠ B' = β' = 127.2487780896° = 127°14'52″ = 0.92106999111 rad
∠ C' = γ' = 105.5044438208° = 105°30'16″ = 1.33001928314 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 = 19 ; ; c = 23 ; ;

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

p = a+b+c = 19+19+23 = 61 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 61 }{ 2 } = 30.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30.5 * (30.5-19)(30.5-19)(30.5-23) } ; ; T = sqrt{ 30252.19 } = 173.93 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 173.93 }{ 19 } = 18.31 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 173.93 }{ 19 } = 18.31 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 173.93 }{ 23 } = 15.12 ; ;

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**2-19**2-23**2 }{ 2 * 19 * 23 } ) = 52° 45'8" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-19**2-23**2 }{ 2 * 19 * 23 } ) = 52° 45'8" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-19**2-19**2 }{ 2 * 19 * 19 } ) = 74° 29'44" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 173.93 }{ 30.5 } = 5.7 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 52° 45'8" } = 11.93 ; ;




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