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: h_a = 22.20436033112
Height: h_b = 11.58444886841
Height: h_c = 11.58444886841

Median: m_a = 22.20436033112
Median: m_b = 14.2921605928
Median: m_c = 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

Calculate another triangle

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

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

6. Inradius

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

7. Circumradius

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

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