22 23 23 triangle

Acute isosceles triangle.

Sides: a = 22   b = 23   c = 23

Area: T = 222.1899108644
Perimeter: p = 68
Semiperimeter: s = 34

Angle ∠ A = α = 57.14437562846° = 57°8'38″ = 0.99773466941 rad
Angle ∠ B = β = 61.42881218577° = 61°25'41″ = 1.07221229797 rad
Angle ∠ C = γ = 61.42881218577° = 61°25'41″ = 1.07221229797 rad

Height: ha = 20.19990098767
Height: hb = 19.3210792056
Height: hc = 19.3210792056

Median: ma = 20.19990098767
Median: mb = 19.34655421222
Median: mc = 19.34655421222

Inradius: r = 6.53549737836
Circumradius: R = 13.09547012559

Vertex coordinates: A[23; 0] B[0; 0] C[10.52217391304; 19.3210792056]
Centroid: CG[11.17439130435; 6.44402640187]
Coordinates of the circumscribed circle: U[11.5; 6.26326832093]
Coordinates of the inscribed circle: I[11; 6.53549737836]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 122.8566243715° = 122°51'22″ = 0.99773466941 rad
∠ B' = β' = 118.5721878142° = 118°34'19″ = 1.07221229797 rad
∠ C' = γ' = 118.5721878142° = 118°34'19″ = 1.07221229797 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 = 22 ; ; b = 23 ; ; c = 23 ; ;

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

p = a+b+c = 22+23+23 = 68 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 68 }{ 2 } = 34 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 34 * (34-22)(34-23)(34-23) } ; ; T = sqrt{ 49368 } = 222.19 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 222.19 }{ 22 } = 20.2 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 222.19 }{ 23 } = 19.32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 222.19 }{ 23 } = 19.32 ; ;

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{ 22**2-23**2-23**2 }{ 2 * 23 * 23 } ) = 57° 8'38" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-22**2-23**2 }{ 2 * 22 * 23 } ) = 61° 25'41" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-22**2-23**2 }{ 2 * 23 * 22 } ) = 61° 25'41" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 222.19 }{ 34 } = 6.53 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 22 }{ 2 * sin 57° 8'38" } = 13.09 ; ;




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