2018 AIME I Problems/Problem 8
Let 
be an equiangular hexagon such that 
, and 
. Denote 
the diameter of the largest circle that fits inside the hexagon. Find 
.
Solutions
Solution Diagram
[asy]
draw((0,0)--(12,20.78)--(24,0)--cycle);
draw((1,1.73)--(2,0));
draw((9,15.59)--(15,15.59));
draw((14,0)--(19,8.66));
label("
",(9,15.59),NW);
label("
",(15,15.59),NE);
label("
",(19,8.66),NE);
label("
",(14,0),S);
label("
",(2,0),S);
label("
",(1,1.73),NW);
pair O;
O=(11.25,7.36);
dot(O);
label("
",O,SW);
draw(Circle(O,6.06));
[/asy]
asymptote code for a picture
- cooljoseph
First of all, draw a good diagram! This is always the key to solving any geometry problem. Once you draw it, realize that 
. Why? Because since the hexagon is equiangular, we can put an equilateral triangle around it, with side length 
. Then, if you drew it to scale, notice that the "widest" this circle can be according to 
is 
. And it will be obvious that the sides won't be inside the circle, so our answer is 
.
-expiLnCalc
