12 13 19 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 13   c = 19

Area: T = 77.0711395472
Perimeter: p = 44
Semiperimeter: s = 22

Angle ∠ A = α = 38.61332185386° = 38°36'48″ = 0.67439277983 rad
Angle ∠ B = β = 42.5376898363° = 42°32'13″ = 0.742240893 rad
Angle ∠ C = γ = 98.85498830984° = 98°51' = 1.72552559253 rad

Height: ha = 12.84552325787
Height: hb = 11.85771377649
Height: hc = 8.11327784707

Median: ma = 15.13327459504
Median: mb = 14.5
Median: mc = 8.1399410298

Inradius: r = 3.50332452487
Circumradius: R = 9.61444619604

Vertex coordinates: A[19; 0] B[0; 0] C[8.84221052632; 8.11327784707]
Centroid: CG[9.28107017544; 2.70442594902]
Coordinates of the circumscribed circle: U[9.5; -1.47991479939]
Coordinates of the inscribed circle: I[9; 3.50332452487]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 141.3876781461° = 141°23'12″ = 0.67439277983 rad
∠ B' = β' = 137.4633101637° = 137°27'47″ = 0.742240893 rad
∠ C' = γ' = 81.15501169016° = 81°9' = 1.72552559253 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 = 12 ; ; b = 13 ; ; c = 19 ; ;

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

p = a+b+c = 12+13+19 = 44 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 44 }{ 2 } = 22 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22 * (22-12)(22-13)(22-19) } ; ; T = sqrt{ 5940 } = 77.07 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 77.07 }{ 12 } = 12.85 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 77.07 }{ 13 } = 11.86 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 77.07 }{ 19 } = 8.11 ; ;

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{ 13**2+19**2-12**2 }{ 2 * 13 * 19 } ) = 38° 36'48" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12**2+19**2-13**2 }{ 2 * 12 * 19 } ) = 42° 32'13" ; ; gamma = 180° - alpha - beta = 180° - 38° 36'48" - 42° 32'13" = 98° 51' ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 77.07 }{ 22 } = 3.5 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 12 }{ 2 * sin 38° 36'48" } = 9.61 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 13**2+2 * 19**2 - 12**2 } }{ 2 } = 15.133 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 19**2+2 * 12**2 - 13**2 } }{ 2 } = 14.5 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 13**2+2 * 12**2 - 19**2 } }{ 2 } = 8.139 ; ;
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