# 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?!

×