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: \(AE \times FA = 3 \times GE \times GF\).


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?!


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