12 23 23 triangle

Acute isosceles triangle.

Sides: a = 12   b = 23   c = 23

Area: T = 133.2221619867
Perimeter: p = 58
Semiperimeter: s = 29

Angle ∠ A = α = 30.24333306298° = 30°14'36″ = 0.52878456963 rad
Angle ∠ B = β = 74.87883346851° = 74°52'42″ = 1.30768734787 rad
Angle ∠ C = γ = 74.87883346851° = 74°52'42″ = 1.30768734787 rad

Height: ha = 22.20436033112
Height: hb = 11.58444886841
Height: hc = 11.58444886841

Median: ma = 22.20436033112
Median: mb = 14.2921605928
Median: mc = 14.2921605928

Inradius: r = 4.59438489609
Circumradius: R = 11.9122480884

Vertex coordinates: A[23; 0] B[0; 0] C[3.13304347826; 11.58444886841]
Centroid: CG[8.71101449275; 3.8611496228]
Coordinates of the circumscribed circle: U[11.5; 3.10876037089]
Coordinates of the inscribed circle: I[6; 4.59438489609]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.757666937° = 149°45'24″ = 0.52878456963 rad
∠ B' = β' = 105.1221665315° = 105°7'18″ = 1.30768734787 rad
∠ C' = γ' = 105.1221665315° = 105°7'18″ = 1.30768734787 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 = 12 ; ; b = 23 ; ; c = 23 ; ;

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

p = a+b+c = 12+23+23 = 58 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 58 }{ 2 } = 29 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29 * (29-12)(29-23)(29-23) } ; ; T = sqrt{ 17748 } = 133.22 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 133.22 }{ 12 } = 22.2 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 133.22 }{ 23 } = 11.58 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 133.22 }{ 23 } = 11.58 ; ;

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{ 23**2+23**2-12**2 }{ 2 * 23 * 23 } ) = 30° 14'36" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12**2+23**2-23**2 }{ 2 * 12 * 23 } ) = 74° 52'42" ; ; gamma = 180° - alpha - beta = 180° - 30° 14'36" - 74° 52'42" = 74° 52'42" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 133.22 }{ 29 } = 4.59 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 12 }{ 2 * sin 30° 14'36" } = 11.91 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 23**2+2 * 23**2 - 12**2 } }{ 2 } = 22.204 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 23**2+2 * 12**2 - 23**2 } }{ 2 } = 14.292 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 23**2+2 * 12**2 - 23**2 } }{ 2 } = 14.292 ; ;
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