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

Question

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

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This triangle is not given to be equilateral or anything else.Only the three altitudes are given which meet at the orthicenter.I do not find any way to find the radius of the circle $DEF.$

Answer 1

The circle through the feet of the altitudes is the nine-point circle, i.e. the circumcircle of the medial triangle. It follows that its circumradius is just $\frac{R}{2}$.