1. Describe a circle passing through two given points, and fulfilling either of the following conditions: 1, touching a given line; 2, touching a given circle. 2. Describe a circle through a given point, and touching two given lines; or touching a given file and a given circle. 3. Describe a circle passing through a given point, having its centre on a given line and touching a given circle. 4. Describe a circle through two given points, and intercepting a given arc on a given circle. 5. A, B, C, D are four collinear points, and EF is a common tangent to the circles described upon AB, CD as diameters: prove that the triangles AEB, CFD are equiangular. 6. The diameter of the circle inscribed in a right-angled triangle is equal to half the sum of the diameters of the circles touching the hypotenuse, the perpendicular from the right angle of the hypotenuse, and the circle described about the right-angled triangle. Questions for Examination on Book III. 1. What is the subject-matter of Book III.? 2. Define equal circles. 3. What is the difference between a chord and a secant? 4. When does a secant become a tangent? 5. What is the difference between a segment of a circle and a sector? 6. What is meant by an angle in a segment? 7. If an arc of a circle be one-sixth of the whole circumference, what is the magnitude of the angle in it? 8. What are linear segments? 9. What is meant by an angle standing on a segment? 10. What are concyclic points? 11. What is a cyclic quadrilateral? 12. How many intersections can a line and a circle have? 13. What does the line become when the points of intersection become consecutive? 14. How many points of intersection can two circles have? 15. What is the reason that if two circles touch they cannot have any other common point? 16. Give one enunciation that will include Propositions xi., xii. of Book III. 17. What Proposition is this a limiting case of? 18. Explain the extended meaning of the word angle. 19. What is Euclid’s limit of an angle? 20. State the relations between Propositions xvi., xviii., xix. 21. What Propositions are these limiting cases of? 22. How many common tangents can two circles have? 23. What is the magnitude of the rectangle of the segments of a chord drawn through a point 3.65 metres distant from the centre of a circle whose radius is 4.25 metres? 24. The radii of two circles are 4.25 and 1.75 feet respectively, and the distance between their centres 6.5 feet; find the lengths of their direct and their transverse common tangents. 25. If a point be h feet outside the circumference of a circle whose diameter is 7920 miles, prove that the length of the tangent drawn from it to the circle is miles. 26. Two parallel chords of a circle are 12 perches and 16 perches respectively, and their distance asunder is 2 perches; find the length of the diameter.