19 21 21 triangle

Acute isosceles triangle.

Sides: a = 19   b = 21   c = 21

Area: T = 177.9199047603
Perimeter: p = 61
Semiperimeter: s = 30.5

Angle ∠ A = α = 53.79330926066° = 53°47'35″ = 0.93988665808 rad
Angle ∠ B = β = 63.10334536967° = 63°6'12″ = 1.10113630364 rad
Angle ∠ C = γ = 63.10334536967° = 63°6'12″ = 1.10113630364 rad

Height: ha = 18.72883208003
Height: hb = 16.94546712003
Height: hc = 16.94546712003

Median: ma = 18.72883208003
Median: mb = 17.0511392905
Median: mc = 17.0511392905

Inradius: r = 5.83334113968
Circumradius: R = 11.77436129336

Vertex coordinates: A[21; 0] B[0; 0] C[8.59552380952; 16.94546712003]
Centroid: CG[9.86550793651; 5.64882237334]
Coordinates of the circumscribed circle: U[10.5; 5.32661582319]
Coordinates of the inscribed circle: I[9.5; 5.83334113968]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 126.2076907393° = 126°12'25″ = 0.93988665808 rad
∠ B' = β' = 116.8976546303° = 116°53'48″ = 1.10113630364 rad
∠ C' = γ' = 116.8976546303° = 116°53'48″ = 1.10113630364 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 = 21 ; ; c = 21 ; ;

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

p = a+b+c = 19+21+21 = 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)(30.5-21)(30.5-21) } ; ; T = sqrt{ 31655.19 } = 177.92 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 177.92 }{ 19 } = 18.73 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 177.92 }{ 21 } = 16.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 177.92 }{ 21 } = 16.94 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 21**2+21**2-19**2 }{ 2 * 21 * 21 } ) = 53° 47'35" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 19**2+21**2-21**2 }{ 2 * 19 * 21 } ) = 63° 6'12" ; ; gamma = 180° - alpha - beta = 180° - 53° 47'35" - 63° 6'12" = 63° 6'12" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 177.92 }{ 30.5 } = 5.83 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 19 }{ 2 * sin 53° 47'35" } = 11.77 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 21**2 - 19**2 } }{ 2 } = 18.728 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 19**2 - 21**2 } }{ 2 } = 17.051 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 19**2 - 21**2 } }{ 2 } = 17.051 ; ;
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