Properties of triangle:
1 A triangle has three sides and three angles:A triangle has three sides and three angles in any ΔABC, we write BC = a, AB = c, AC = b
And ÐBAC = ÐA , ÐABC = ÐB , ÐACB = ÐC
2 In ΔABC :
(i) A+B+C = p
(ii) a+b > c, b+c > a, c+a > b
(iii) a > 0 , b > 0 , c > 0
3 Sine formula :
(a/sinA) = (b/sinB) = (c/sinC) = k(say)
Or (sinA/a) = (sinB/b) = (sinC/c) = k(say)
4 Cosine formula :
5 Projection formula :
a = b cos C + ccos B
b = b c cos A + a cos C
c = a cos B + b cos A
6 Napier’s Analogies :
7 Half angled formula – In any ΔABC :
Where 2s = a + b + c
8 Δ , Area of triangle :
(i) Δ = 1/2 ab sin C = 1/2 bc sin A 1/2 ca sin B
(ii) 
10 Circumcircle of triangle and its radius :
(i) R = (a/2sinA) = (b/2sinB) = (c/sinC)
(ii) R = (abc/4Δ) where R is circumradius
11 Incircle of a triangle and its radius :
(iii) r = Δ / s
(iv) r = (s-a) tan (A/2) = (s-b) tan (B/2) = (s-c) tan (C/2)
(v) r = 4R sin (A/2) sin (B/2) sin (C/2)
(vi) cos A + cos B + cos C = 1 + (r/R)
12 The radius of the escribed circles are given by :
(i) r1 = (Δ / ( s-1)) , r2 = (Δ / ( s-b) ) , r3 = (Δ / (s-c))
(ii) r1 = s tan (A/2) , r2 = s tan (B/2) , r3 = s tan (C/2)
(iii) r1 = 4R sin (A/2) cos (B/2) cos (C/2)
r2 = 4R cos (A/2) sin (B/2) cos (C/2)
r3 = 4R cos (A/2) cos (B/2) sin (C/2)
(iv) r1 + r2 + r3 - r = 4R
(v) (1/r1) + (1/r2) + (1/r3) = (1/r)
(vii) (1/bc) + (1/ca) + (1/ ab) = (1/2Rr)
(viii) r1r2 + r2r3 + r3r1 = s2
(ix) Δ = 2R2 sin A sin B sin C = 4Rr cos (A/2) cos (B/2) cos (C/2)
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