23. [12] Regular hexagon ABCDEF has side length 2. Circle ω lies inside the hexagon and is tangent to segments AB and AF . There exist two perpendicular lines tangent to ω that pass through C and E , respectively. Given that these two lines do not intersect on line AD , compute the radius of ω . Proposed by: Karthik Venkata Vedula √ √ 3 3 − 3 3 Answer: = ( 3 − 1) 2 2 Solution 1: A F B O P E C D Let O be the center of ω , and let the two tangent lines intersect at P . Note that O lies on the external angle bisector of ∠ CP E because the tangents are symmetric about line P O . Additionally, O lies on ◦ the perpendicular bisector of CE by symmetry. By Fact 5, COP E is cyclic and ∠ COE = 90 . To √ ◦ finish, observe that ∠ COD = 45 . Dropping the altitude CH down to AD gives OH = CH = 3. √ √ √ 3 3 3 − 3 So, AO = AH − OH = 3 − 3. The desired answer is then · AO = . 2 2 ◦ Solution 2: Another way to get ∠ COE = 90 is as follows. Let ω meet the tangents from C and E at Q and R , respectively. Observe OC = OE (as O lies on the

◦ ∼ perpendicular bisector of CE ) and OQ = OR , so △ OCQ △ OER . Then ∠ COE = ∠ QOR = 90 .