Hexy Circles
is one side of a regular hexagon. It has length 1. is chosen so that the two small circles are congruent. If the radius, , of the larger circle is , where are co-prime positive integers and are square-free, submit .
The answer is 21.
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1 solution
Answer
= 5 + 7 + 6 + 3
= 21
Extrapolated FG will meet extrapolated AB at the same point AB meets DC (by congruent triangles from congruent incircles). Let this point be X. AB = BX = 1 so AX = 2. If we put an equilateral triangle of side 2 under the hexagon with one of its edge being AX, it will be a similar triangle to the one containing the big incircle which radius we're looking for with resizing factor of 3. Obviously the sides of this big triangle are 2 and 3, and the other side of √7 can be found using the angle of 60°.