A problem by Fahim Shahriar Shakkhor

Level pending

In $ABC$ equilateral triangle the length of each side is $24$ . $D$ is the midpoint of $BC$ . $DE$ and $DF$ are drawn perpendicular to $AB$ and $AC$ respectively from the point $D$ . $O$ is the intersection point of $AD$ and $EF$ . Find the circum-radius of $\triangle AOF$ .

The answer is 9.

1 solution

Whatever 21
Feb 9, 2014

Because it's an equilateral triangle, EF and CB are parallel.

We know angle ACD= 60 , so 24x sin 60= AD .

AD=12\sqrt{3}

Perpendicular AD bisects angle A , so angle CAD =30.

Triangle AFD is a right angled triangle. So AF= AD. cos 30

AF=18

As the triangle AOF is a right angled triangle, we can say that AF is the diameter of the circumcircle of triangle AOF. So circum radius= 9.

A problem by Fahim Shahriar Shakkhor

The answer is 9.

1 solution

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