Exercises.
1. Another solution may be inferred from Proposition viii., Cor. 2.
2. If through any point within a circle the chord be drawn, which is bisected in that point, its half is a mean proportional between the segments of any other chord passing through the same point.
3. The tangent to a circle from any external point is a mean proportional between the segments of any secant passing through the same point.
4. If through the middle point C of any arc of a circle any secant be drawn cutting the chord of the arc in D, and the circle again in E, the chord of half the arc is a mean proportional between CD and CE.
5. If a circle be described touching another circle internally and two parallel chords, the perpendicular from the centre of the former on the diameter of the latter, which bisects the chords, is a mean proportional between the two extremes of the three segments into which the diameter is divided by the chords.
6. If a circle be described touching a semicircle and its diameter, the diameter of the circle is a harmonic mean between the segments into which the diameter of the semicircle is divided at the point of contact.
7. State and prove the Proposition corresponding to Ex. 5, for external contact of the circles.
PROP. XIV.—Theorem.
