# Inspired by Calvin Lin 2 : An open challenge to all of Brilliant

**Geometry**Level 5

Suppose in the previous problem \(BC\) may not be \(10\) i.e. there is no information about it. We only have the condition that was mentioned: \(AE \times FA = 3 \times GE \times GF\).

Then:

1) Is it possible to have: A fool-proof compasses & straightedge (which is a "scale" without graduations, i.e. basically a straight rod that cannot measure) only, construction of the \(\Delta ABC\) simply given length \(AB\) and the circumcircle of the \(\Delta ABC\). (of course \(AB < 2\times R\))
In other words, on a sheet of paper, I draw a circle out and beside it somewhere a line segment (of lesser length than the circle's diameter) and ask you to construct a triangle \(ABC\) with the above property (\(AE \times FA = 3 \times GE \times GF\)) within the circle (the circle should pass through \(A,B,C)\) with the length of \(AB\) equal to the segment's length. Can you *always* do it?

2) Should the triangle be necessarily equilateral?

I am more interested in the solution! Do tell me the how or the how not and the why or the why not?!