elem.6.11
给定两条线段，求作第三条线段，使得第一条与第二条的比等于第二条与第三条的比。
求 AB 与 AC 的第三比例项：以 A 为顶点作两条线 AB、AC，在 AB 延长线上取 BD=AC，连 BC，过 D 作平行于 BC 的直线交 AC 延长线于 E，则 AB:AC=AC:CE。
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 BA, AC be the two given straight lines, and let them be placed so as to contain any angle; thus it is required to find a third proportional to BA, AC.
设给定线段为BA和AC，将它们放置成任意角，并延长至D和E，使BD等于AC。
For let them be produced to the points D, E, and let BD be made equal to AC; [I. 3] let BC be joined, and through D let DE be drawn parallel to it.
连接BC，过D作DE平行于BC。
[I. 31] Since, then, BC has been drawn parallel to DE, one of the sides of the triangle ADE, proportionally, as AB is to BD, so is AC to CE.
由于在三角形ADE中，BC平行于DE，因此AB比BD等于AC比CE。
[VI. 2] But BD is equal to AC; therefore, as AB is to AC, so is AC to CE.
因为BD等于AC，所以AB比AC等于AC比CE，即CE为所求的第三比例项。