elem.3.32
若一直线与圆相切，且从切点作一直线在圆内截圆，则该直线与切线所成角等于圆中相对弓形内的角。
本页以“切线与弦所成角等于相对弓形内角”整体图解辅助阅读；点、线、角、圆索引已按命题文字和证明步骤校订，可与证明和问答联动。
正文图形由校订坐标生成；点、线、角、圆可与证明和问答联动。
For let a straight line EF touch the circle ABCD at the point B, and from the point B let there be drawn across, in the circle ABCD, a straight line BD cutting it; I say that the angles which BD makes with the tangent EF will be equal to the angles in the alternate segments of the circle, that is, that the angle FBD is equal to the angle constructed in the segment BAD, and the angle EBD is equal to the angle constructed in the segment DCB. For let BA be drawn from B at right angles to EF, let a point C be taken at random on the circumference BD, and let AD, DC, CB be joined. Then, since a straight line EF touches the circle ABCD at B, and BA has been drawn from the point of contact at right angles to the tangent, the centre of the circle ABCD is on BA.
设直线EF切圆ABCD于点B，从B作线段BD截圆。
[III. 19] Therefore BA is a diameter of the circle ABCD; therefore the angle ADB, being an angle in a semicircle, is right. [III. 31] Therefore the remaining angles BAD, ABD are equal to one right angle.
过B作BA垂直于EF，则BA为直径，故角ADB为直角。
[I. 32] But the angle ABF is also right; therefore the angle ABF is equal to the angles BAD, ABD. Let the angle ABD be subtracted from each; therefore the angle DBF which remains is equal to the angle BAD in the alternate segment of the circle.
角ABF为直角，等于角BAD与角ABD之和；减去角ABD，得角DBF等于角BAD。
Next, since ABCD is a quadrilateral in a circle, its opposite angles are equal to two right angles. [III. 22] But the angles DBF, DBE are also equal to two right angles; therefore the angles DBF, DBE are equal to the angles BAD, BCD, of which the angle BAD was proved equal to the angle DBF; therefore the angle DBE which remains is equal to the angle DCB in the alternate segment DCB of the circle.
四边形ABCD内接于圆，对角和等于两直角；角DBF与角DBE和也为两直角，减去已证相等的角，得角DBE等于角DCB。