Given two circles about the same centre, to inscribe in the greater circle an equilateral polygon with an even number of sides which does not touch the lesser circle.
给定两个同心圆，在大圆内作一个边数为偶数的等边多边形，使其不与小圆相切。
两同心圆，公共圆心 K：外圆 ABCD、内圆 EFGH。直径 BKD 与内圆右交于 G；过 G 作 GA⊥BD，延长至 C，则 AC 切内圆于 G。在外圆弧 BAD 上反复平分得点 L，使 LD 比 AD 短；过 L 作 LM⊥BD，延长至 N，得 LD=DN，且 LN∥AC。
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Let ABCD, EFGH be the two given circles about the same centre K; thus it is required to inscribe in the greater circle ABCD an equilateral polygon with an even number of sides which does not touch the circle EFGH. For let the straight line BKD be drawn through the centre K, and from the point G let GA be drawn at right angles to the straight line BD and carried through to C; therefore AC touches the circle EFGH.
设两同心圆为ABCD和EFGH，圆心为K。作直径BKD，过G作AC垂直于BD并延长至C，则AC切小圆于G。
[III. 16, Por.] Then, bisecting the circumference BAD, bisecting the half of it, and doing this continually, we shall leave a circumference less than AD.
将弧BAD不断平分，最终得到一段弧LD小于AD。过L作LM垂直于BD并延长至N，连接LD和DN，则LD等于DN。
[X. 1] Let such be left, and let it be LD; from L let LM be drawn perpendicular to BD and carried through to N, and let LD, DN be joined; therefore LD is equal to DN.
由于LN平行于AC，而AC切小圆，故LN不与小圆相切，因此LD和DN更远离小圆。
[III. 3, I. 4] Now, since LN is parallel to AC, and AC touches the circle EFGH, therefore LN does not touch the circle EFGH; therefore LD, DN are far from touching the circle EFGH.
以LD为边在大圆内作等边多边形，其边数为偶数，且所有边均不与小圆相切，即为所求。