Question

In a triangle $ABC;AD,BE$ and $CF$ are the altitudes and $R$ is the circumradius, then the radius of the circle $DEF$ is

• A. $2R$
• B. $R$
• C. $\frac{R}{2}$
• D. none of these.

Answer 1

The correct option is C $\frac{R}{2}$

In pedal $△DEF$
$EF=a\cos A$
$DE=c\cos C$
$DF=b\cos B$
If circum-radius of $△DEF$ is $R_1$
Then, $R_1=\frac{\left(a\cos A\right)\left(b\cos B\right)\left(c\cos C\right)}{4\times \frac{1}{2}DF.DE\sin \left(\angle EDF\right)}$
Here $\angle EDF={180}^0-2A$
$=\frac{abc\cos A\cos B\cos C}{4\times \frac{1}{2}\times b\cos B.c\cos C.\sin ({180}^0-2A)}$
$=\frac{a\cos A}{2\sin 2A}$
Substitute for $a=2R\sin A$
$=\frac{2R\sin A\cos A}{4\sin A\cos A}$
$\therefore R_1=\frac{R}{2}$