10 18 23 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 18   c = 23

Area: T = 86.08768021244
Perimeter: p = 51
Semiperimeter: s = 25.5

Angle ∠ A = α = 24.57546375825° = 24°34'29″ = 0.42989083383 rad
Angle ∠ B = β = 48.46875991175° = 48°28'3″ = 0.84659191851 rad
Angle ∠ C = γ = 106.95877633° = 106°57'28″ = 1.86767651302 rad

Height: ha = 17.21773604249
Height: hb = 9.5655200236
Height: hc = 7.48658088804

Median: ma = 20.03774649095
Median: mb = 15.28107067899
Median: mc = 8.93302855497

Inradius: r = 3.37659530245
Circumradius: R = 12.02327488356

Vertex coordinates: A[23; 0] B[0; 0] C[6.63304347826; 7.48658088804]
Centroid: CG[9.87768115942; 2.49552696268]
Coordinates of the circumscribed circle: U[11.5; -3.5076635077]
Coordinates of the inscribed circle: I[7.5; 3.37659530245]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 155.4255362417° = 155°25'31″ = 0.42989083383 rad
∠ B' = β' = 131.5322400883° = 131°31'57″ = 0.84659191851 rad
∠ C' = γ' = 73.04222367° = 73°2'32″ = 1.86767651302 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 = 10 ; ; b = 18 ; ; c = 23 ; ;

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

p = a+b+c = 10+18+23 = 51 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 51 }{ 2 } = 25.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25.5 * (25.5-10)(25.5-18)(25.5-23) } ; ; T = sqrt{ 7410.94 } = 86.09 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 86.09 }{ 10 } = 17.22 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 86.09 }{ 18 } = 9.57 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 86.09 }{ 23 } = 7.49 ; ;

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{ 18**2+23**2-10**2 }{ 2 * 18 * 23 } ) = 24° 34'29" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10**2+23**2-18**2 }{ 2 * 10 * 23 } ) = 48° 28'3" ; ; gamma = 180° - alpha - beta = 180° - 24° 34'29" - 48° 28'3" = 106° 57'28" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 86.09 }{ 25.5 } = 3.38 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 10 }{ 2 * sin 24° 34'29" } = 12.02 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 18**2+2 * 23**2 - 10**2 } }{ 2 } = 20.037 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 23**2+2 * 10**2 - 18**2 } }{ 2 } = 15.281 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 18**2+2 * 10**2 - 23**2 } }{ 2 } = 8.93 ; ;
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