[Sidenote: If a strait line falling upon the circumference of a circle be produced till it reach the semidiameter, and that part of it, which is intercepted between the circumference and the semidiameter, be equal to that part of the semidiameter which is between the point of concourse and the centre, the reflected line will be parallel to the semidiameter.] 9. If a strait line, falling upon the circumference of a circle, be produced till it reach the semidiameter, and that part of it which is intercepted between the circumference and the semidiameter be equal to that part of the semidiameter which is between the point of concourse and the centre, the reflected line will be parallel to the semidiameter. Let any line A B (in the 9th figure) be the semidiameter of the circle whose centre is A; and upon the circumference B D let the strait line C D fall, and be produced till it cut A B in E, so that E D and E A may be equal; and from the incident line C D let the line D F be reflected. I say, A B and D F will be parallel. Let A G be drawn through the point D. Seeing, therefore, E D and E A are equal, the angles E D A and E A D will also be equal. But the angles F D G and E D A are equal; for each of them is half the angle E D H or F D C. Wherefore the angles F D G and E A D are equal; and consequently D F and A B are parallel; which was to be proved. Coroll. If E A be greater then E D, then D F and A B being produced will concur; but if E A be less than E D, then B A and D H being produced will concur. [Sidenote: If from a point within a circle two strait lines be drawn to the circumference, and their reflected lines meet in the circumference of the same circle, the angle made by the reflected lines will be a third part of the angle made by the incident lines.] 10. If from a point within a circle two strait lines be drawn to the circumference, and their reflected lines meet in the circumference of the same circle, the angle made by the lines of reflection will be a third part of the angle made by the lines of incidence. From the point B (in the 10th figure) taken within the circle whose centre is A, let the two strait lines B C and B D be drawn to the circumference; and let their reflected lines C E and D E meet in the circumference of the same circle at the point E. I say, the angle C E D will be a third part of the angle C B D.