Properties of Triangle

1. Some results of conditional identity :
If A + B + C = π then
i. Sin2A + Sin2B + Sin2C = 4SinA.SinB.SinC
ii. SinA + SinB + SinC = 4CosA/2.CosB/2.CosC/2
iii. CosA + CosB + CosC = 1 + (4SinA/2.SinB/2.SinC/2)
iv. Sin²A + Sin²B + Sin²C = 2 + 2CosA.CosB.CosC
v. Cos2A/2 + Cos2B/2 + Cos2C/2 = 2 + 2SinA/2.SinB/2.SinC/2
vi. tanA + tanB + tanC = tanA.tanB.tanC
vii. tanA/2.tanB/2. + tanB/2.tanC/2 + tanC/2.tanA/2 = 1
viii. CotA/2.CotB/2 + CotB/2.CotC/2 + CotC/2.CotA/2 = 1

2.     Sine Law :

In any ΔABC
a/SinA = b/SinB = c/SinC = 2R (Radius of Circumference circle)
a = 2RSinA, SinA = a/2R
b = 2RSinB, SinB = b/2R
c = 2RSinC, SinC = c/2R

3.     Cosine Law :
In any Δ ABC,
CosA = [(b² + c² – a²)/2bc]
CosB = [(c² + a² – b²)/2ac]
CosC = [(a² + b² – c²)/2ab]

4. Projection law :
a = bcosA + cCosB
b = cCosA + aCosC
c = aCosA + bCosA

5. Tangent Law :
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)]

6. Half angle formulae :
Here, S = (a + b + c)/2
SinA = √[{(s-b)(s-c)}/{bc}]
Sin(B/2) = √[{(s-a)(s-c)}/{ac}]
Sin(C/2) = √[{(s-a)(s-b)}/{ab}]
Cos(A/2) = √[{s(s-a)}/{bc}]
Cos(B/2) = √[{s(s-b)}/{ac}]
Cos(C/2) = √[{s(s-c)}/{ab}]
Tan(A/2) = √[{(s-b)(s-c)}/{s(s-a)}]
Tan(B/2) = √[{(s-a)(s-c)}/{s(s-b)}]
Tan(C/2) = √[{(s-a)(s-b)}/{s(s-c)}]
Cot(A/2) = √[{s(s-a)}/{(s-b)(s-c)}]
Cot(B/2) = √[{s(s-b)}/{(s-a)(s-c)}]
Cot(C/2) = √[{s(s-c)}/{(s-b)(s-a)}]

7. Area of triangle :
i. Δ =1/2.abSinC,
ii. Δ = √{s(s-a)(s-b)(s-c)}
iii. Δ = ¼√[2a²b² + 2b²c² + 2a²c² – a⁴ – b⁴ – c⁴]
iv. Δ = {abc}/{4R}
v. Δ = ½b.h
vi. Δ = √3/4.a2(for equilateral Δ)
Also,
tan(A/2)= [{(s-b)(s-c)}/Δ]= [Δ/{s(s-a)}]
Cot(A/2)= [{s(s-a)}/Δ]

8.     Formula for radii,

r →radius of in-circle.
i. r = Δ/s
ii. r = (s-a)tan(A/2)= (s-b)tan(B/2) = (s-c)tan(C/2)
iii. r = 4Rsin(A/2).sin(B/2).sin(C/2)

 

If r₁, r₂ & r₃ represents radius of ex-circles opposite to A, B, C then,
i.r₁ =∆/(s-a) r₂ = ∆/(s-b) r₃ = ∆/(s-c)
ii. r₁ = stan(A/2), r₂ = stan(B/2), r₃ = stan(C/2)
iii.r₁ = 4Rsin(A/2).cos(B/2).cos(C/2)
r₂ = 4Rcos(A/2).sin(B/2).cos(C/2)
r₃ = 4Rcos(A/2).cos(B/2).sin(C/2)

iv.r₁ = asec(A/2).cos(B/2).cos(C/2)
r₂ = acos(A/2).sec(B/2).cos(C/2)
r₃ = acos(A/2).cos(B/2).sec(C/2)

some important results are :
i. 4R = r₁+ r₂+ r₂- r
ii. 1/r₂ + 1/r₂ + 1/ r₃ = 1/r
iii. rr₁r₂r₃ = Δ²
iv. cosA + cosB + cosC = 1 + r/R
v. In an equilateral triangle R. r = 1/6.(side of the Δ²)
vi. If sinA, sinB, sinC are in A.P./G.P./H.P. then a, b, c are in A.P./G.P./H.P. respectively.
vii. If a, b, c are in H.P. then SinA, SinB, SinC are in A.P.
viii. If r₁, r₂, r₃ are in H.P. then sinA, sinB, sinC are in A.P.
ix. If cot(A/2), cot(B/2), cot(C/2)are in A.P. then a, b, c are in A.P.
x. If cotA, cotB, cotC are in A.P. then a², b², c² are in A.P.