Let A and B be different points on a circle with center O. With only a straightedge and a compass, can you construct a straight line through O meeting the segment AB at C, C strictly between A and B, and meeting the circle at D, so that C is between O and D and the segments AC and AD have the same length?

Since hasn't been stated where A and B are, but cannot be colinear with O, set the compas to an equal radius to the circle. Make a major arc from both A and B so that the arcs intersect twice. Label these E and F. Then, using the straightedge, connect A to B. Run a line through E and F such that it crosses the circle at each end. The intersection of line AB and line EF is C, and C is between O and D on line EF. For segment AC to equal AD, however is not possible because angle ACD is a right angle, making triangle ACD a right triangle with segment AD as hypotenuse. It cannot be equal to a leg.

Posted by Bret on 2005-07-20 17:21:19