[Sidenote: If two strait lines drawn from the same point without a circle fall upon the circumference, and the lines reflected from them being produced meet within the circle, they will make an angle equal to twice that angle, which is made by two strait lines drawn from the centre to the points of incidence, together with the angle which the incident lines themselves make.] 4. If two strait lines drawn from the same point without a circle fall upon the circumference, and the lines reflected from them being produced meet within the circle, they will make an angle equal to twice that angle, which is made by two strait lines drawn from the centre to the points of incidence, together with the angle which the incident lines themselves make. Let the two strait lines A B and A C (in fig. 4) be drawn from the point A to the circumference of the circle, whose centre is D; and let the lines reflected from them be B E and C G, and, being produced, make within the circle the angle H; also let the two strait lines D B and D C be drawn from the centre D to the points of incidence B and C. I say, the angle H is equal to twice the angle at D together with the angle at A. For let A C be produced howsoever to I. Therefore the angle I C H, which is external to the triangle C K H, will be equal to the two angles C K H and C H K. Again, the angle I C D, which is external to the triangle C L D, will be equal to the two angles C L D and C D L. But the angle I C H is double to the angle I C D, and is therefore equal to the angles C L D and C D L twice taken. Wherefore the angles C K H and C H K are equal to the angles C L D and C D L twice taken. But the angle C L D, being external to the triangle A L B, is equal to the two angles L A B and L B A; and consequently C L D twice taken is equal to L A B and L B A twice taken. Wherefore C K H and C H K are equal to the angle C D L together with L A B and L B A twice taken. Also the angle C K H is equal to the angle L A B once and A B K, that is, L B A twice taken. Wherefore the angle C H K is equal to the remaining angle C D L, that is, to the angle at D, twice taken, and the angle L A B, that is, the angle at A, once taken; which was to be proved.