11. If a tangent be made the diameter of a circle, whose centre is the point of contact, a strait line drawn from the centre of the former circle to the centre of the latter circle, will make two angles with the tangent, that is, with the diameter of the latter circle, equal to two right angles, by the last article. And because, by the [6th article], the tangent has on both sides equal inclination to the circle, each of them will be a right angle; as also the semidiameter will be perpendicular to the same tangent. Moreover, the semidiameter, inasmuch as it is the semidiameter, is the least strait line which can be drawn from the centre to the tangent; and every other strait line, that reaches the tangent, will pass out of the circle, and will therefore be greater than the semidiameter. In like manner, of all the strait lines, which may be drawn from the centre to the tangent, that is the greatest which makes the greatest angle with the perpendicular; which will be manifest, if about the same centre another circle be described, whose semidiameter is a strait line taken nearer to the perpendicular, and there be drawn a perpendicular, that is, a tangent, to the same.
From whence it is also manifest, that if two strait lines, which make equal angles on either side of the perpendicular, be produced to the tangent, they will be equal.
The general definition of parallels; the properties of strait parallels.
12. There is in Euclid a definition of strait-lined parallels; but I do not find that parallels in general are anywhere defined; and therefore for an universal definition of them, I say that any two lines whatsoever, strait or crooked, as also any two superficies, are PARALLEL; when two equal strait lines, wheresoever they fall upon them, make always equal angles with each of them.
From which definition it follows; first, that any two strait lines, not inclined opposite ways, falling upon two other strait lines, which are parallel, and intercepting equal parts in both of them, are themselves also equal and parallel. As if A B and C D (in the [third figure]), inclined both the same way, fall upon the parallels A C and B D, and A C and B D be equal, A B and C D will also be equal and parallel. For the perpendiculars B E and D F being drawn, the right angles E B D and F D H will be equal. Wherefore, seeing E F and B D are parallel, the angles E B A and F D C will be equal. Now if D C be not equal to B A, let any other strait line equal to B A be drawn from the point D; which, seeing it cannot fall upon the point C, let it fall upon G. Wherefore A G will be either greater or less than B D; and therefore the angles E B A and F D C are not equal, as was supposed. Wherefore A B and C D are equal; which is the first.
Again, because they make equal angles with the perpendiculars B E and D F; therefore the angle C D H will be equal to the angle A B D, and, by the definition of parallels, A B and C D will be parallel; which is the second.
That plane, which is included both ways within parallel lines, is called a PARALLELOGRAM.
Coroll. I. From this last it follows, that the angles A B D and C D H are equal, that is, that a strait line, as B H, falling upon two parallels, as A B and C D, makes the internal angle A B D equal to the external and opposite angle C D H.
Coroll. II. And from hence again it follows, that a strait line falling upon two parallels, makes the alternate angles equal, that is, the angle A G F, in the [fourth figure], equal to the angle G F D. For seeing G F D is equal to the external opposite angle E G B, it will be also equal to its vertical angle A G F, which is alternate to G F D.
Coroll. III. That the internal angles on the same side of the line F G are equal to two right angles. For the angles at F, namely, G F C and G F D, are equal to two right angles. But G F D is equal to its alternate angle A G F. Wherefore both the angles G F C and A G F, which are internal on the same side of the line F G, are equal to two right angles.