**Basic Triangle Formulas & Identities**

Topics Discussed:

Area of Triangle = ½ (Any Side) × (Corresponding Altitude)

**Heron’s Formula**

∆ =√(s(s – a)(s – b)(s – c))

S = ½ (a + b + c)…….. Semi-Perimeter

Where a, b, c……sides of a ∆

**Area of an Equilateral Triangle (∆)**

Area= (√3a²)/4

**Centroid (G) of a Triangle**

G = ({x_{1}+ x_{2}+ x_{3} /3}; {y_{1 }+ y_{2 }+ y_{3}} /3 )

Area = ½ [x_{1}(y_{2 }– y_{3}) + x_{2}(y_{3 }– y_{1}) + x_{3}(y_{1 }– y_{2})]

**Length of Medians of a Triangle**

L_{1} = ½ (√(2b² + 2c² – a²) )

L_{2 }= ½ (√(2c² + 2a² – b²) )

L3 = ½ (√(2a² + 2b² – c²)

4(L_{1}^{2} + L_{2}^{2 }+ L_{3}^{2}) = 3(a² + b² + c²)

**Length of Angle Bisector (x)**

x = (2bc/ (b + c)) cos (A/2)

**Sine Rule**

a/sin A = b/sin B = c/sin C = 2R

Where R is radius of Circumcircle

**Cosine Rule**

Cos A = (b² + c² – a²)/2bc

Cos B = (c² + a² – b²)/2ca

Cos C = (a² + b² – c²)/2ab

**Area (Δ)**

∆ = 1/2 ab sin C = 1/2 bc sin A = 1/2 ca sin B

∆ = abc/4R

r = ∆ /s

Where R = Circumcircle Radius

r = Incircle radius

**Projection Formula**

a = b cos C + c cos B

b = c cos A + a cos C

c = a cos B + b cos A

**Tangent Rule**

tan ((B – C)/2) = (b – c)/(b + c) cot (A/2)

tan ((C – A)/2 ) = (c – a)/(c + a) cot (B/2)

tan ((A – B)/2) = (a – b)/(a + b) cot (C/2)

**M-N Theorem**

(m + n) cot θ = m cot α – n cot β

(m + n) cot θ = m cot B – n cot C

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