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(A) Let the circumradius of $	riangle DEF$ be $R'$.In $	riangle ABC$, the pedal triangle $	riangle DEF$ has angles $\angle FDE = 180^{\circ} - 2A$,

Vedclass · Accepted Answer

$\frac{R}{2}$

Vedclass · Answer

(A) Let the circumradius of $	riangle DEF$ be $R'$.In $	riangle ABC$, the pedal triangle $	riangle DEF$ has angles $\angle FDE = 180^{\circ} - 2A$, $\angle DEF = 180^{\circ} - 2B$, and $\angle EFD = 180^{\circ} - 2C$.The side length $EF$ of the pedal triangle is given by $EF = R \sin 2A$.Using the sine rule in $	riangle DEF$, we have $2R' = \frac{EF}{\sin(\angle FDE)}$.Substituting the values, $2R' = \frac{R \sin 2A}{\sin(180^{\circ} - 2A)} = \frac{R \sin 2A}{\sin 2A} = R$.Therefore, $R' = \frac{R}{2}$.

If in a $\triangle ABC$, $AD$, $BE$, and $CF$ are the altitudes and $R$ is the circumradius of $\triangle ABC$, then the radius of the circumcircle of $\triangle DEF$ is

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