In the circle EFGH' inscribe the square EFGH. [IV. 6.
This square shall be greater than half of the circle EFGH.
For the square EFGH is half of the square which can be formed by drawing straight lines to touch the circle at the points E, F, G, H;
and the square thus formed is greater than the circle;
therefore the square EFGH is greater than half of the circle.
Bisect the arcs EF, FG, GH, HE at the points K,L,M,N;
and join EK, KF, FL, LG, GM, MH, HN, NE. Then each of the triangles EKF, FLG, GMH, HNE shall be greater than half of the segment of the circle in which it stands.
For the triangle EKF is half of the parallelogram which can be formed by drawing a straight line to touch the circle at K, and parallel straight lines through E and F, and the parallelogram thus formed is greater than the segment FEK; therefore the triangle EKF is greater than half of the segment.
And similarly for the other triangles.
Therefore the sum of all these triangles is together greater than half of the sum of the segments of the circle in which they stand.
Again, bisect EK, KF, &c. and form triangles as before;
then the sum of these triangles is greater than half of the sum of the segments of the circle in which they stand.
