An exam for high school students had the following problem:

Let the point EE be the midpoint of the line segment ADAD on the square ABCDABCD. Then let a circle be determined by the points EE, BB and CC as shown on the diagram. Which of the geometric figures has the greater perimeter, the square or the circle?

Of course, there are some ways to solve this problem. One method is as follows: assume the side lengths of the square is 11, put everything somewhere on a Cartesian coordinate system, find the midpoint of the circle using the coordinates of EE, BB and CC, then find the radius of the circle, and finally use the radius to calculate the circle’s circumference and compare it to the perimeter of the square.

The problem with that method is that ostensibly this problem is supposed to be very simple; it shouldn’t require the student to know the formula for the midpoint of a circle given three coordinates. Therefore the question here is: **does there exist a simple way to solve the problem without knowing any complicated geometric formulas?**

This is also “computational” and not purely geometic. Let the side of the square be 11. Let OO be the centre of the circle and GG be the middle of the segement CB¯¯¯¯¯¯¯¯CB¯; then r=|OB|r=|OB| is the radius of the circle and 1−r=|OG|1−r=|OG|. By Pythagoras we have (1−r)2+(12)2=|OG|2+|GB|2=|OB|2=r2(1−r)2+(12)2=|OG|2+|GB|2=|OB|2=r2 from which one gets r=58r=58. Again one gets the same as in the answer by abel. PS I also would like to see a purely geometric argument.