20 23 25 triangle

Acute scalene triangle.

Sides: a = 20   b = 23   c = 25

Area: T = 217.0810630181
Perimeter: p = 68
Semiperimeter: s = 34

Angle ∠ A = α = 49.03108802499° = 49°1'51″ = 0.85657502955 rad
Angle ∠ B = β = 60.26442868924° = 60°15'51″ = 1.05218102276 rad
Angle ∠ C = γ = 70.70548328576° = 70°42'17″ = 1.23440321304 rad

Height: ha = 21.70880630182
Height: hb = 18.87765765375
Height: hc = 17.36664504145

Median: ma = 21.84403296678
Median: mb = 19.5
Median: mc = 17.55770498661

Inradius: r = 6.38547244171
Circumradius: R = 13.24439269114

Vertex coordinates: A[25; 0] B[0; 0] C[9.92; 17.36664504145]
Centroid: CG[11.64; 5.78988168048]
Coordinates of the circumscribed circle: U[12.5; 4.37662541098]
Coordinates of the inscribed circle: I[11; 6.38547244171]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130.969911975° = 130°58'9″ = 0.85657502955 rad
∠ B' = β' = 119.7365713108° = 119°44'9″ = 1.05218102276 rad
∠ C' = γ' = 109.2955167142° = 109°17'43″ = 1.23440321304 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 = 20 ; ; b = 23 ; ; c = 25 ; ;

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

p = a+b+c = 20+23+25 = 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 * (34-20)(34-23)(34-25) } ; ; T = sqrt{ 47124 } = 217.08 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 217.08 }{ 20 } = 21.71 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 217.08 }{ 23 } = 18.88 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 217.08 }{ 25 } = 17.37 ; ;

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+25**2-20**2 }{ 2 * 23 * 25 } ) = 49° 1'51" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 20**2+25**2-23**2 }{ 2 * 20 * 25 } ) = 60° 15'51" ; ; gamma = 180° - alpha - beta = 180° - 49° 1'51" - 60° 15'51" = 70° 42'17" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 217.08 }{ 34 } = 6.38 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 20 }{ 2 * sin 49° 1'51" } = 13.24 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 23**2+2 * 25**2 - 20**2 } }{ 2 } = 21.84 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 25**2+2 * 20**2 - 23**2 } }{ 2 } = 19.5 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 23**2+2 * 20**2 - 25**2 } }{ 2 } = 17.557 ; ;
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