elem.6.13
给定两条线段，求作它们的比例中项。
在 AC 上以 AC 为直径作半圆；B 把 AC 分成 AB 与 BC，过 B 作 AC 的垂线交半圆于 D，则 BD 是 AB、BC 的比例中项。
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 AB, BC be the two given straight lines; thus it is required to find a mean proportional to AB, BC.
设AB、BC为两条给定线段，将它们放置在同一直线上，以AC为直径作半圆ADC。
Let them be placed in a straight line, and let the semicircle ADC be described on AC; let BD be drawn from the point B at right angles to the straight line AC, and let AD, DC be joined.
从点B作AC的垂线BD，连接AD、DC。
Since the angle ADC is an angle in a semicircle, it is right.
因为半圆上的角ADC是直角（III.31），所以在直角三角形ADC中，DB是从直角顶点到底边的垂线。
[III. 31] And, since, in the right-angled triangle ADC, DB has been drawn from the right angle perpendicular to the base, therefore DB is a mean proportional between the segments of the base, AB, BC.
根据VI.8的推论，DB是底边两段AB、BC的比例中项。