Object > Path > Add Anchor Points will add a point in the middle of a straight path made with one segment. Then you can select the added point and press the button with the scissor on the control bar.
edit 1: Another way is if you check Show Center button in the Attributes panel, then using the Scissor tool with the Smart Guides turned on will show the "center" label and snap to it when the mouse is over the center of the path.
edit 2: Forget the second suggestion, I didn't check it all the way, when I clicked with the scissor to try cutting the path, the warning popped out showing that Illustrator thinks the center point is an end anchor point. Very poor programming
and disappointingly Add Anchor Point tool also doesn't work on the center point.
Or, to make it independent of the accurate sizing and positioning of a rectangle, using the Smart Guides, you may:
1) Select the path and cut it with the scissors Tool where it crosses the inner circle (Smart Guides say intersect;
2) With the short part selected and with the right Reference Point ticked in the Transform pallete, add /2 to the W box and press enter;
3) Join the halved path and the rest of the original path.
In 3), the newest versions should let you do it without further ado, in older versions, you may lock/hide the inner circle and drag over the coinciding points.
In any case, you may delete the joined Anchor Point if desired.