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Properties of Triangle

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                 PROPERTIES OF TRIANGLES

Parts of a triangle

We know that a triangle has six parts viz. Three angles and three sides.
In any triangle ABC, the measures of the angles BAC, CBA and ACB are denoted respectively by the letters A, B and C, and the sides BC,CA and AB opposite to these angles are represented by a, b and c respectively. The radius of the circumcircle, i.e. Circumradius will be denoted by R and the area of the triangle of \vartriangle \!\, . All the six parts of the triangle are not independent; they satisfy certain relations.
In geometry you are familier with the following relations :
(i)      A + B + C = 180 degree     (ii) a+b > c          b + c > a         c + a > b


Law of sines or sine rule

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

Law of cosine or cosine rule

1.      a^2 = b^2 + c^2 -2bc cosA     or         cos A = (b^2 + c^2 – a^2) / 2bc
2.      b^2 = c^2 + a^2 – 2ca cos B   or         cos B = (c^2 + a^2 – b^2) / 2ca
3.      c^2 = a^2 + b^2 – 2ab cos C  or         cos C = (a^2 + b^2 – c^2) / 2ab

Law of tangents or tangent rule
1.      tan (B-C)/2 = (b-c)/(b+c)  cot A/2
2.      tan (C-A)/2 = (c-a)/(c+a)  cot B/2
3.      tan (A-B)/2 = (a-b)/(a+b)  cot C/2


The relation between the three sides and two angles of any triangle ABC  are as follows
1.      a = b cos C + c cos B
2.      b = c cos A +a cos C
3.      c = a cos B + b cos A
To find sines of half an angle of a triangle  in terms of the sides
1.      sin A/2 = \sqrt{\ } \!\, (s-b)(s-c)/bc
2.      sin B/2 = \sqrt{\ } \!\, (s-c)(s-a)/ca
3.      sin C/2 = \sqrt{\ } \!\, (s-a)(s-b)/ab

To find cosines of half an angle of a triangle in terms of the sides
1.      cos A/2 = \sqrt{\ } \!\, s(s-a)/bc
2.      cos B/2 = \sqrt{\ } \!\, s(s-b)/ca
3.      cos C/2 = \sqrt{\ } \!\, s(s-c)/ab

To find tangent of half an angle of triangle in terms of the sides
1.      tan A/2 = \sqrt{\ } \!\, (s-b)(s-c) / s(s-a)
2.      tan B/2 = \sqrt{\ } \!\, (s-c)(s-a) / s(s-b)
3.      tan C/2 = \sqrt{\ } \!\, (s-a)(s-b) / s(s-c)

Expressions for area of a triangle ABC
1.      Area = 1/2 ab sin C
2.      Area = 1/2 bc sin A
3.      Area = 1/2 ca sin B

Values of  sines of angles in terms  of sides and area
1.      sin A = 2/bc \sqrt{\ } \!\,s(s-a)(s-b)(s-c)
2.      sin B = 2/ca \sqrt{\ } \!\,s(s-a)(s-b)(s-c)
3.      sin C = 2/ab \sqrt{\ } \!\,s(s-a)(s-b)(s-c)

Values of tangents  and cotangents of half  an angle in terms of sides and area
1.      tan A/2 = (s-b)(s-c) / \sqrt{\ } \!\,s(s-a)(s-b)(s-c)
2.      tan B/2 = (s-c)(s-a) / \sqrt{\ } \!\,s(s-a)(s-b)(s-c)
3.      tan C/2 = (s-a)(s-b) / \sqrt{\ } \!\,s(s-a)(s-b)(s-c)
4.      cot A/2 = s(s-a) / \sqrt{\ } \!\,s(s-a)(s-b)(s-c)
5.      cot B/2 = s(s-b) / \sqrt{\ } \!\,s(s-a)(s-b)(s-c)
6.      cot C/2 = s(s-c) / \sqrt{\ } \!\,s(s-a)(s-b)(s-c)

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