In acute-angled triangles the square on the side subtending the acute angle is less than the squares on the sides containing the acute angle by twice the rectangle contained by one of the sides about the acute angle, namely that on which the perpendicular falls, and the straight line cut off within by the perpendicular towards the acutc angle.
在锐角三角形中，锐角所对边上的正方形，小于夹该角两边上正方形之和，差量等于两倍某边与投影段所成矩形。
锐角三角形 ABC，三角内 B 为锐角；过 A 作 AD⊥BC，垂足 D 落在 BC 内部。AC² = AB² + BC² − 2·BC·BD。
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 ABC be an acute-angled triangle having the angle at B acute, and let AD be drawn from the point A perpendicular to BC; I say that the square on AC is less than the squares on CB, BA by twice the rectangle contained by CB, BD.
在锐角三角形中，从一顶点向对边作垂线，垂足落在边内。
For, since the straight line CB has been cut at random at D, the squares on CB, BD are equal to twice the rectangle contained by CB, BD and the square on DC.
相关边被分成两段，较大平方关系要从整段中减去投影造成的矩形。
[II. 7] Let the square on DA be added to each; therefore the squares on CB, BD, DA are equal to twice the rectangle contained by CB, BD and the squares on AD, DC.
使用 euclid-elements/book2-prop-007 或前面分解式整理平方。
But the square on AB is equal to the squares on BD, DA, for the angle at D is right; [I. 47] and the square on AC is equal to the squares on AD, DC; therefore the squares on CB, BA are equal to the square on AC and twice the rectangle CB, BD, so that the square on AC alone is less than the squares on CB, BA by twice the rectangle contained by CB, BD.
再用 euclid-elements/book1-prop-047 消去垂线平方，得到差量为两倍矩形。