13 19 19 triangle

Acute isosceles triangle.

Sides: a = 13   b = 19   c = 19

Area: T = 116.0488211964
Perimeter: p = 51
Semiperimeter: s = 25.5

Angle ∠ A = α = 40.01103801763° = 40°37″ = 0.69883128691 rad
Angle ∠ B = β = 69.99548099118° = 69°59'41″ = 1.22216398923 rad
Angle ∠ C = γ = 69.99548099118° = 69°59'41″ = 1.22216398923 rad

Height: ha = 17.85435710714
Height: hb = 12.21656012593
Height: hc = 12.21656012593

Median: ma = 17.85435710714
Median: mb = 13.21993040664
Median: mc = 13.21993040664

Inradius: r = 4.55109102731
Circumradius: R = 10.11100222067

Vertex coordinates: A[19; 0] B[0; 0] C[4.44773684211; 12.21656012593]
Centroid: CG[7.81657894737; 4.07218670864]
Coordinates of the circumscribed circle: U[9.5; 3.45986918075]
Coordinates of the inscribed circle: I[6.5; 4.55109102731]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.9989619824° = 139°59'23″ = 0.69883128691 rad
∠ B' = β' = 110.0055190088° = 110°19″ = 1.22216398923 rad
∠ C' = γ' = 110.0055190088° = 110°19″ = 1.22216398923 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 = 13 ; ; b = 19 ; ; c = 19 ; ;

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

p = a+b+c = 13+19+19 = 51 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 51 }{ 2 } = 25.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25.5 * (25.5-13)(25.5-19)(25.5-19) } ; ; T = sqrt{ 13467.19 } = 116.05 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 116.05 }{ 13 } = 17.85 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 116.05 }{ 19 } = 12.22 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 116.05 }{ 19 } = 12.22 ; ;

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{ 13**2-19**2-19**2 }{ 2 * 19 * 19 } ) = 40° 37" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-13**2-19**2 }{ 2 * 13 * 19 } ) = 69° 59'41" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 19**2-13**2-19**2 }{ 2 * 19 * 13 } ) = 69° 59'41" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 116.05 }{ 25.5 } = 4.55 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 40° 37" } = 10.11 ; ;




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