19 23 23 triangle

Acute isosceles triangle.

Sides: a = 19   b = 23   c = 23

Area: T = 198.9990420624
Perimeter: p = 65
Semiperimeter: s = 32.5

Angle ∠ A = α = 48.79223305655° = 48°47'32″ = 0.85215868181 rad
Angle ∠ B = β = 65.60438347173° = 65°36'14″ = 1.14550029178 rad
Angle ∠ C = γ = 65.60438347173° = 65°36'14″ = 1.14550029178 rad

Height: ha = 20.94663600657
Height: hb = 17.30435148368
Height: hc = 17.30435148368

Median: ma = 20.94663600657
Median: mb = 17.68547391838
Median: mc = 17.68547391838

Inradius: r = 6.1232782173
Circumradius: R = 12.62774922789

Vertex coordinates: A[23; 0] B[0; 0] C[7.8487826087; 17.30435148368]
Centroid: CG[10.28326086957; 5.76878382789]
Coordinates of the circumscribed circle: U[11.5; 5.21657033326]
Coordinates of the inscribed circle: I[9.5; 6.1232782173]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 131.2087669435° = 131°12'28″ = 0.85215868181 rad
∠ B' = β' = 114.3966165283° = 114°23'46″ = 1.14550029178 rad
∠ C' = γ' = 114.3966165283° = 114°23'46″ = 1.14550029178 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 = 19 ; ; b = 23 ; ; c = 23 ; ;

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

p = a+b+c = 19+23+23 = 65 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 65 }{ 2 } = 32.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 32.5 * (32.5-19)(32.5-23)(32.5-23) } ; ; T = sqrt{ 39597.19 } = 198.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 198.99 }{ 19 } = 20.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 198.99 }{ 23 } = 17.3 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 198.99 }{ 23 } = 17.3 ; ;

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{ 19**2-23**2-23**2 }{ 2 * 23 * 23 } ) = 48° 47'32" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-19**2-23**2 }{ 2 * 19 * 23 } ) = 65° 36'14" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-19**2-23**2 }{ 2 * 23 * 19 } ) = 65° 36'14" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 198.99 }{ 32.5 } = 6.12 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 48° 47'32" } = 12.63 ; ;




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