Each set of points {A,B,C}, {D,E,F} and {G,H,I} are equilaterally spaced and lie respectively upon the circumferences of 3 concentric circles centred at O.

The points D, A, B & I are collineal in that order as are F, C, A & H and E, B, C & G.

Scalene triangles GDB, HEC and IFA are formed by these points and contain Δ ABC in their overlapping.


Note: This is not scaled.
Area "X" is that which is bounded by the scalene triangles (pink) while Area "Y" is all other areas (white) within the largest circle.

If line segment AB has a length of 1 unit and segment AI is 3 units:
1. What is the value of segment AF when the Pink region fills exactly half of the largest circle?
2. What is the proportion, Pink:White (or X:Y), when / AFI is a right angle? [I have considered X, the pink region, to equal 1 in my solution].

AF= 2.7749

Posted by K Sengupta on 2023-03-27 22:39:44