Bibliotheca Polyglotta

PROPOSITION 31.

THEOR. 27. PROPOS. 31.

第三十一題

In a circle the angle in the semicircle is right, that in a greater segment less than a right angle, and that in a less segment greater than a right angle; and further the angle of the greater segment is greater than a right angle, and the angle of the less segment less than a right angle.

IN circulo angulus, qui in semicirculo, rectus est: qui autem in maiore segmento, minor recto: qui vero in minore segmento, maior est recto. Et insuper angulus maioris segmenti, recto quidem maior est: minoris autem segmenti angulus, minor est recto.

五支
負半圜角、必直角。負大分角、小於直角。負小分角、大於直角。大圜分角、大於直角。小圜分角、小於直角。

Let ABCD be a circle, let BC be its diameter, and E its centre, and let BA, AC, AD, DC be joined; I say that the angle BAC in the semicircle BAC is right, the angle ABC in the segment ABC greater than the semicircle is less than a right angle, and the angle ADC in the segment ADC less than the semicircle is greater than a right angle.

Let AE be joined, and let BA be carried through to F.

Then, since BE is equal to EA, the angle ABE is also equal to the angle BAE. [I. 5] Again, since CE is equal to EA, the angle ACE is also equal to the angle CAE. [I. 5] Therefore the whole angle BAC is equal to the two angles ABC, ACB. But the angle FAC exterior to the triangle ABC is also equal to the two angles ABC, ACB; [I. 32] therefore the angle BAC is also equal to the angle FAC; therefore each is right; [I. Def. 10] therefore the angle BAC in the semicircle BAC is right.

Next, since in the triangle ABC the two angles ABC, BAC are less than two right angles, [I. 17] and the angle BAC is a right angle, the angle ABC is less than a right angle; and it is the angle in the segment ABC greater than the semicircle.

Next, since ABCD is a quadrilateral in a circle, and the opposite angles of quadrilaterals in circles are equal to two right angles, [III. 22] while the angle ABC is less than a right angle, therefore the angle ADC which remains is greater than a right angle; and it is the angle in the segment ADC less than the semicircle.

I say further that the angle of the greater segment, namely that contained by the circumference ABC and the straight line AC, is greater than a right angle; and the angle of the less segment, namely that contained by the circumference ADC and the straight line AC, is less than a right angle. This is at once manifest. For, since the angle contained by the straight lines BA, AC is right, the angle contained by the circumference ABC and the straight line AC is greater than a right angle. Again, since the angle contained by the straight lines AC, AF is right, the angle contained by the straight line CA and the circumference ADC is less than a right angle.

Therefore etc. Q. E. D.

一系。凡角形之內。一角與兩角幷、等。其一角必直角。何者。其外角與內相對之兩角等。則與外角等之內交角。豈非直角。
二系。大分之角。大於直角。小分之角。小於直角。終無有角等於直角。又從小過大。從大過小。非大卽小。終無相等。係此題四五論、甚明。與本篇十六題增注、互相發也。

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