JEE Mathematics Properties of Triangles Notes Set 01

Sides, Angles and Area of a Triangle

Sine rule: \( \dfrac a{\sin A}=\dfrac b{\sin B}=\dfrac c{\sin C}=2R \). Cosine rule: \( a^2=b^2+c^2-2bc\cos A \), i.e. \( \cos A=\dfrac{b^2+c^2-a^2}{2bc} \). Projection rule: \( a=b\cos C+c\cos B \).
With \( s=\dfrac{a+b+c}2 \): area \( \Delta=\tfrac12bc\sin A=\sqrt{s(s-a)(s-b)(s-c)}=rs=\dfrac{abc}{4R} \). The altitude on side a is \( h_a=\dfrac{2\Delta}a \).

Half-Angle Formulae

\( \sin\dfrac A2=\sqrt{\dfrac{(s-b)(s-c)}{bc}} \), \( \cos\dfrac A2=\sqrt{\dfrac{s(s-a)}{bc}} \), \( \tan\dfrac A2=\sqrt{\dfrac{(s-b)(s-c)}{s(s-a)}}=\dfrac{\Delta}{s(s-a)} \).
Useful consequences: \( \cot\dfrac B2\cot\dfrac C2=\dfrac s{s-a} \), and \( \prod\sin\dfrac A2=\dfrac r{4R} \), \( \prod\cos\dfrac A2=\dfrac s{4R} \).

Inradius, Circumradius and Exradii

Inradius \( r=\dfrac\Delta s=4R\sin\dfrac A2\sin\dfrac B2\sin\dfrac C2 \). Exradius opposite A: \( r_1=\dfrac\Delta{s-a}=4R\sin\dfrac A2\cos\dfrac B2\cos\dfrac C2 \), and similarly \( r_2,r_3 \).
Relations: \( r_1+r_2+r_3=4R+r \), \( \dfrac1{r_1}+\dfrac1{r_2}+\dfrac1{r_3}=\dfrac1r \), \( r_1r_2+r_2r_3+r_3r_1=s^2 \), \( r_1r_2r_3=\Delta s \), and \( r_1-r=4R\sin^2\dfrac A2 \). Euler's inequality: \( R\ge2r \), with equality for an equilateral triangle.

Special Points and Triangles Related to a Triangle

Orthocentre H: \( \angle BHC=180^\circ-A \) and the circumradius of triangle BHC equals R. Incentre I: \( \angle BIC=90^\circ+\dfrac A2 \), and I divides the bisector from A in the ratio \( (b+c):a \). Centroid G divides each median in the ratio 2:1.
The orthic (pedal) triangle has sides \( a\cos A,\ b\cos B,\ c\cos C \) and perimeter \( \dfrac{2\Delta}R \). The brocard angle \( \omega \) satisfies \( \cot\omega=\cot A+\cot B+\cot C=\dfrac{a^2+b^2+c^2}{4\Delta} \).

Quick Tests for the Type of Triangle

If a relation forces equality in an inequality (for example \( \sin A=\dfrac{b+c}{2\sqrt{bc}}\ge1 \)), the triangle must be of a special type, here right and isosceles.
Right angle at A: \( b^2+c^2=a^2 \), \( r=s-a \), \( R=\dfrac a2 \). Equilateral: \( R=2r \). To find the number of integral values of a side, let the included angle vary between its limits and use the cosine rule.