elem.4.9
给定一个正方形，求作一个圆外切于该正方形。
正方形 ABCD；连接对角线 AC、BD 交于中心 E；以 E 为心、EA 为半径作圆，过四顶点即为外接圆。
正文图形由校订坐标生成；点、线、角、圆可与证明和问答联动。
Let ABCD be the given square; thus it is required to circumscribe a circle about the square ABCD. For let AC, BD be joined, and let them cut one another at E. Then, since DA is equal to AB, and AC is common, therefore the two sides DA, AC are equal to the two sides BA, AC; and the base DC is equal to the base BC; therefore the angle DAC is equal to the angle BAC.
连接对角线AC和BD，设交点为E。
[I. 8] Therefore the angle DAB is bisected by AC. Similarly we can prove that each of the angles ABC, BCD, CDA is bisected by the straight lines AC, DB.
由三角形全等（边边边）得AC平分角DAB，同理可证AC和DB平分各内角。
Now, since the angle DAB is equal to the angle ABC, and the angle EAB is half the angle DAB, and the angle EBA half the angle ABC, therefore the angle EAB is also equal to the angle EBA; so that the side EA is also equal to EB. [I. 6] Similarly we can prove that each of the straight lines EA, EB is equal to each of the straight lines EC, ED. Therefore the four straight lines EA, EB, EC, ED are equal to one another.
由角EAB等于角EBA，得EA等于EB；同理可证EA、EB、EC、ED彼此相等。
Therefore the circle described with centre E and distance one of the straight lines EA, EB, EC, ED will pass also through the remaining points; and it will have been circumscribed about the square ABCD. Let it be circumscribed, as ABCD.
以E为圆心、EA为半径作圆，必过B、C、D，即为所求外接圆。