20 23 23 triangle

Acute isosceles triangle.

Sides: a = 20   b = 23   c = 23

Area: T = 207.1233151772
Perimeter: p = 66
Semiperimeter: s = 33

Angle ∠ A = α = 51.54329234811° = 51°32'35″ = 0.98995937208 rad
Angle ∠ B = β = 64.22985382594° = 64°13'43″ = 1.12109994664 rad
Angle ∠ C = γ = 64.22985382594° = 64°13'43″ = 1.12109994664 rad

Height: ha = 20.71223151772
Height: hb = 18.01107088497
Height: hc = 18.01107088497

Median: ma = 20.71223151772
Median: mb = 18.22877261336
Median: mc = 18.22877261336

Inradius: r = 6.27664591446
Circumradius: R = 12.77701803365

Vertex coordinates: A[23; 0] B[0; 0] C[8.69656521739; 18.01107088497]
Centroid: CG[10.56552173913; 6.00435696166]
Coordinates of the circumscribed circle: U[11.5; 5.55222523202]
Coordinates of the inscribed circle: I[10; 6.27664591446]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 128.4577076519° = 128°27'25″ = 0.98995937208 rad
∠ B' = β' = 115.7711461741° = 115°46'17″ = 1.12109994664 rad
∠ C' = γ' = 115.7711461741° = 115°46'17″ = 1.12109994664 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 = 20 ; ; b = 23 ; ; c = 23 ; ;

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

p = a+b+c = 20+23+23 = 66 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 66 }{ 2 } = 33 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33 * (33-20)(33-23)(33-23) } ; ; T = sqrt{ 42900 } = 207.12 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 207.12 }{ 20 } = 20.71 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 207.12 }{ 23 } = 18.01 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 207.12 }{ 23 } = 18.01 ; ;

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{ 20**2-23**2-23**2 }{ 2 * 23 * 23 } ) = 51° 32'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-20**2-23**2 }{ 2 * 20 * 23 } ) = 64° 13'43" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-20**2-23**2 }{ 2 * 23 * 20 } ) = 64° 13'43" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 207.12 }{ 33 } = 6.28 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20 }{ 2 * sin 51° 32'35" } = 12.77 ; ;




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