21 23 26 triangle

Acute scalene triangle.

Sides: a = 21   b = 23   c = 26

Area: T = 230.0433474152
Perimeter: p = 70
Semiperimeter: s = 35

Angle ∠ A = α = 50.29879022662° = 50°17'52″ = 0.87878640014 rad
Angle ∠ B = β = 57.42110296072° = 57°25'16″ = 1.00221860265 rad
Angle ∠ C = γ = 72.28110681266° = 72°16'52″ = 1.26215426257 rad

Height: ha = 21.90989023002
Height: hb = 20.00437803611
Height: hc = 17.69656518579

Median: ma = 22.18767077323
Median: mb = 20.64658228221
Median: mc = 17.77663888346

Inradius: r = 6.57326706901
Circumradius: R = 13.64774203912

Vertex coordinates: A[26; 0] B[0; 0] C[11.30876923077; 17.69656518579]
Centroid: CG[12.43658974359; 5.89985506193]
Coordinates of the circumscribed circle: U[13; 4.15435627277]
Coordinates of the inscribed circle: I[12; 6.57326706901]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 129.7022097734° = 129°42'8″ = 0.87878640014 rad
∠ B' = β' = 122.5798970393° = 122°34'44″ = 1.00221860265 rad
∠ C' = γ' = 107.7198931873° = 107°43'8″ = 1.26215426257 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 = 21 ; ; b = 23 ; ; c = 26 ; ;

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

p = a+b+c = 21+23+26 = 70 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 70 }{ 2 } = 35 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 35 * (35-21)(35-23)(35-26) } ; ; T = sqrt{ 52920 } = 230.04 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 230.04 }{ 21 } = 21.91 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 230.04 }{ 23 } = 20 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 230.04 }{ 26 } = 17.7 ; ;

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{ 21**2-23**2-26**2 }{ 2 * 23 * 26 } ) = 50° 17'52" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-21**2-26**2 }{ 2 * 21 * 26 } ) = 57° 25'16" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 26**2-21**2-23**2 }{ 2 * 23 * 21 } ) = 72° 16'52" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 230.04 }{ 35 } = 6.57 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 50° 17'52" } = 13.65 ; ;




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