20 20 23 triangle

Acute isosceles triangle.

Sides: a = 20   b = 20   c = 23

Area: T = 188.1755283977
Perimeter: p = 63
Semiperimeter: s = 31.5

Angle ∠ A = α = 54.99003678046° = 54°54'1″ = 0.95881921787 rad
Angle ∠ B = β = 54.99003678046° = 54°54'1″ = 0.95881921787 rad
Angle ∠ C = γ = 70.19992643908° = 70°11'57″ = 1.22552082961 rad

Height: ha = 18.81875283977
Height: hb = 18.81875283977
Height: hc = 16.3633068172

Median: ma = 19.0921883092
Median: mb = 19.0921883092
Median: mc = 16.3633068172

Inradius: r = 5.9743818539
Circumradius: R = 12.2232646627

Vertex coordinates: A[23; 0] B[0; 0] C[11.5; 16.3633068172]
Centroid: CG[11.5; 5.45443560573]
Coordinates of the circumscribed circle: U[11.5; 4.14404215449]
Coordinates of the inscribed circle: I[11.5; 5.9743818539]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 125.1099632195° = 125°5'59″ = 0.95881921787 rad
∠ B' = β' = 125.1099632195° = 125°5'59″ = 0.95881921787 rad
∠ C' = γ' = 109.8010735609° = 109°48'3″ = 1.22552082961 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 = 20 ; ; c = 23 ; ;

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

p = a+b+c = 20+20+23 = 63 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 63 }{ 2 } = 31.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31.5 * (31.5-20)(31.5-20)(31.5-23) } ; ; T = sqrt{ 35409.94 } = 188.18 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 188.18 }{ 20 } = 18.82 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 188.18 }{ 20 } = 18.82 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 188.18 }{ 23 } = 16.36 ; ;

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{ 20**2+23**2-20**2 }{ 2 * 20 * 23 } ) = 54° 54'1" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 20**2+23**2-20**2 }{ 2 * 20 * 23 } ) = 54° 54'1" ; ; gamma = 180° - alpha - beta = 180° - 54° 54'1" - 54° 54'1" = 70° 11'57" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 188.18 }{ 31.5 } = 5.97 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 20 }{ 2 * sin 54° 54'1" } = 12.22 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 20**2+2 * 23**2 - 20**2 } }{ 2 } = 19.092 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 23**2+2 * 20**2 - 20**2 } }{ 2 } = 19.092 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 20**2+2 * 20**2 - 23**2 } }{ 2 } = 16.363 ; ;
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