16 23 23 triangle

Acute isosceles triangle.

Sides: a = 16   b = 23   c = 23

Area: T = 172.5110869223
Perimeter: p = 62
Semiperimeter: s = 31

Angle ∠ A = α = 40.70988127965° = 40°42'32″ = 0.71105028179 rad
Angle ∠ B = β = 69.64655936018° = 69°38'44″ = 1.21655449179 rad
Angle ∠ C = γ = 69.64655936018° = 69°38'44″ = 1.21655449179 rad

Height: ha = 21.56438586528
Height: hb = 15.00109451498
Height: hc = 15.00109451498

Median: ma = 21.56438586528
Median: mb = 16.13222658049
Median: mc = 16.13222658049

Inradius: r = 5.56548667491
Circumradius: R = 12.26658937929

Vertex coordinates: A[23; 0] B[0; 0] C[5.56552173913; 15.00109451498]
Centroid: CG[9.52217391304; 55.0003150499]
Coordinates of the circumscribed circle: U[11.5; 4.2666397841]
Coordinates of the inscribed circle: I[8; 5.56548667491]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.2911187204° = 139°17'28″ = 0.71105028179 rad
∠ B' = β' = 110.3544406398° = 110°21'16″ = 1.21655449179 rad
∠ C' = γ' = 110.3544406398° = 110°21'16″ = 1.21655449179 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 = 16 ; ; b = 23 ; ; c = 23 ; ;

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

p = a+b+c = 16+23+23 = 62 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 62 }{ 2 } = 31 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31 * (31-16)(31-23)(31-23) } ; ; T = sqrt{ 29760 } = 172.51 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 172.51 }{ 16 } = 21.56 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 172.51 }{ 23 } = 15 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 172.51 }{ 23 } = 15 ; ;

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{ 16**2-23**2-23**2 }{ 2 * 23 * 23 } ) = 40° 42'32" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-16**2-23**2 }{ 2 * 16 * 23 } ) = 69° 38'44" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-16**2-23**2 }{ 2 * 23 * 16 } ) = 69° 38'44" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 172.51 }{ 31 } = 5.56 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 16 }{ 2 * sin 40° 42'32" } = 12.27 ; ;




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