# JEE Main & Advanced Mathematics Trigonometric Equations Pedal Triangle

## Pedal Triangle

Category : JEE Main & Advanced

Let the perpendiculars AD, BE and CF from the vertices A, B and C on the opposite sides BC, CA and AB of $\Delta ABC$ respectively, meet at O. Then O is the orthocentre of the $\Delta ABC$. The triangle DEF is called the pedal triangle of the $\Delta ABC$.

Othocentre of the triangle is the incentre of the pedal triangle.

If O is the orthocentre and DEF the pedal triangle of the $\Delta ABC$, where AD, BE, CF are the perpendiculars drawn from A, B, C on the opposite sides BC, CA, AB respectively, then

(i)   $OA=2R\cos A,OB=2R\cos B$and $OC=2R\cos C$

(ii) $OD=2R\cos B\cos C,OE=2R\cos C\cos A$ and $OF=2R\cos A\cos B$

(1) Sides and angles of a pedal triangle: The angles of pedal triangle DEF are: $180-2A,\,180-2B,\,180-2C$ and sides of pedal triangle are:

$EF=a\cos A$ or $R\sin 2A$; $FD=b\cos B$ or $R\sin 2B$; $DE=c\cos C$ or $R\sin 2C$

If given $\Delta ABC$ is obtuse, then angles are represented by $2A,$ $2B$, $2C-{{180}^{o}}$ and the sides are $a\cos A,\,\,b\cos B,\,\,-\,\,c\cos C$.

(2) Area and circum-radius and in-radius of pedal triangle : Area of pedal triangle $=\frac{1}{2}(\text{Product of the sides)}\times$ (sine of included angle)

$\Delta =\frac{1}{2}{{R}^{2}}.\sin 2A.\sin 2B.\sin 2C$

Circum-radius of pedal triangle$=\frac{EF}{2\sin FDE}=\frac{R\sin 2A}{2\sin ({{180}^{o}}-2A)}=\frac{R}{2}$

In-radius of pedal triangle $=\frac{\text{area of }\Delta DEF}{\text{semi-perimeter of }\Delta DEF}$

$=\frac{\frac{1}{2}{{R}^{2}}\sin 2A.\sin 2B.\sin 2C}{2R\sin A.\sin B.\sin C}=2R\cos A.\cos B.\cos C$

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