Math and Logic

Cut Exactly in Half

How can we draw a single line to divide a circle (or group of circles) precisely in half? If we're successful, we should have the same area and circumference on both sides of the line.

With a single circle, we can do this by drawing any line through the center, but what about when we have more circles?

What about two circles touching side by side? The previous strategy works, and we gain a new one as well. We can make a line that passes through the circles' centers, or we can pass a line between the two circles. Either of these will work.

But there's an additional way to think about two circles that includes both of the above and many other values besides. Imagine drawing a rectangle around these two circles and labelling the center of the rectangle. Because any line drawn through the center will divide the two halves of the rectangle symmetrically, it will also divide the circles into two parts with equal area and perimeter.

Another way to describe this is that a line through the two circles' point of tangency will evenly divide their areas and circumferences.

How can these ideas help with the problem below?

Today's Challenge

Is it possible to position the line so that it evenly divides these five circles into two parts of equal area and perimeter?


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