# Properties of Triangles

• Tips:
• Sine rule \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}
• Area of the triangle (Δ) = \frac{1}{2} bc · sinA = \frac{1}{2} ca · sinB = \frac{1}{2} ab · sinC
• Napier's analogy. In any triangle ABC = Tan\left(\frac{B-C}{2}\right)=\frac{b-c}{b+c}\ \cot\frac{A}{2}
• In any triangle \frac{a\cdot \sin(B-C)}{b^{2}-c^{2}}=\frac{b\cdot \sin(C-A)}{c^{2}-a^{2}}=\frac{c\cdot \sin(A-B)}{a^{2}-b^{2}}
• In any triangle (b-c)\cot\frac{A}{2}+(c-a)\cot\frac{B}{2}+(a-b)\cot\frac{c}{2} = 0
• In any triangle a cos A + b cos B + c cos C = 2a sinB · sin C.
• In any triangle a2 = b2 + c2 − 2bc.cos A.
• \sin \frac{A}{2}=\sqrt{\frac{(s-b)(s-c)}{bc}}
• r=(s-a)Tan \frac{A}{2}=(s-b)Tan \frac{B}{2}=(s-c)Tan \frac{C}{2}
• r_{1}=S \ Tan \frac{A}{2};r_{2}= S \ Tan \frac{B}{2};r_{3}=S \ Tan \frac{C}{2}
• r=4R \ \sin \frac{A}{2}\cdot\ \sin \frac{B}{2}\cdot\ \sin \frac{C}{2}

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1. Sine Rule  \tt \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}=\frac{1}{2R}, where R is the radius of the circumcircle of the ΔABC.

2. Cosine Rule
\tt \cos A=\frac{b^{2}+c^{2}-a^{2}}{2bc},
\tt \cos B=\frac{a^{2}+c^{2}-b^{2}}{2ac}
and \tt \cos C=\frac{a^{2}+b^{2}-c^{2}}{2ab}

3. Projection Rule  a = b cos C + c cos B,
b = c cos A + a cos C and c = a cos B + b cos A

4. Napier's Analogy
\tt \tan \frac{B-C}{2}=\frac{b-c}{b+c} \cot \frac{A}{2},
\tt \tan \frac{C-A}{2}=\frac{c-a}{c+a} \cot \frac{B}{2} \ and \tan \frac{A-B}{2}=\frac{a-b}{a+b} \cot \frac{C}{2}

5. Area of Triangles
Consider a triangle of side a, b and c.
a). \tt \Delta=\frac{1}{2}bc \ sin \ A=\frac{1}{2}ca \ sin \ B = \frac{1}{2}ab \ sin \ C

6. \tt \Delta=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)} where, s =\frac{a+b+c}{2}(semi-perimeter of triangle)

7. Trigonometrical Ratios of Half of the Angles
a). \tt sin\frac{A}{2}=\sqrt{\frac{\left(s-b\right)\left(s-c\right)}{bc}}
\tt sin\frac{B}{2}=\sqrt{\frac{\left(s-c\right)\left(s-a\right)}{ac}}
\tt sin\frac{C}{2}=\sqrt{\frac{\left(s-a\right)\left(s-b\right)}{ab}}
b).\tt cos\frac{A}{2}=\sqrt{\frac{s\left(s-a\right)}{bc}}
\tt cos\frac{B}{2}=\sqrt{\frac{s\left(s-b\right)}{ac}}
\tt cos\frac{C}{2}=\sqrt{\frac{s\left(s-c\right)}{ab}}
c). \tt tan\frac{A}{2}=\sqrt{\frac{\left(s-b\right)\left(s-c\right)}{s\left(s-a\right)}}
\tt tan\frac{B}{2}=\sqrt{\frac{\left(s-a\right)\left(s-c\right)}{s\left(s-b\right)}}
\tt tan\frac{C}{2}=\sqrt{\frac{\left(s-a\right)\left(s-b\right)}{s\left(s-c\right)}}