27. What is the locus of the centres of all circles touching a given circle in a given point? 28. What is the condition that must be fulfilled that four points may be concyclic? 29. If the angle in a segment of a circle be a right angle and a-half, what part of the whole circumference is it? 30. Mention the converse Propositions of Book III. which are proved directly. 31. What is the locus of the middle points of equal chords in a circle? 32. The radii of two circles are 6 and 8, and the distance between their centres 10; find the length of their common chord. 33. If a figure of any even number of sides be inscribed in a circle, prove that the sum of one set of alternate angles is equal to the sum of the remaining angles. Exercises on Book III. 1. If two chords of a circle intersect at right angles, the sum of the squares on their segments is equal to the square on the diameter. 2. If a chord of a given circle subtend a right angle at a fixed point, the rectangle of the perpendiculars on it from the fixed point and from the centre of the given circle is constant. Also the sum of the squares of perpendiculars on it from two other fixed points (which may be found) is constant. 3. If through either of the points of intersection of two equal circles any line be drawn meeting them again in two points, these points are equally distant from the other intersection of the circles. 4. Draw a tangent to a given circle so that the triangle formed by it and two fixed tangents to the circle shall be—1, a maximum; 2, a minimum. 5. If through the points of intersection A, B of two circles any two lines ACD, BEF be drawn parallel to each other, and meeting the circles again in C, D, E, F; then CD = EF. 6. In every triangle the bisector of the greatest angle is the least of the three bisectors of the angles. 7. The circles whose diameters are the four sides of any cyclic quadrilateral intersect again in four concyclic points. 8. The four angular points of a cyclic quadrilateral determine four triangles whose orthocentres (the intersections of their perpendiculars) form an equal quadrilateral. 9. If through one of the points of intersection of two circles we draw two common chords, the lines joining the extremities of these chords make a given angle with each other. 10. The square on the perpendicular from any point in the circumference of a circle, on the chord of contact of two tangents, is equal to the rectangle of the perpendiculars from the same point on the tangents. 11. Find a point in the circumference of a given circle, the sum of the squares on whose distances from two given points may be a maximum or a minimum. 12. Four circles are described on the sides of a quadrilateral as diameters. The common chord of any two on adjacent sides is parallel to the common chord of the remaining two.