2 BOOK OF MATHEMATICAL TKOBLEMS. 



1 and the part produced together with the square on the 

 given line, "will be equal to four times the square on the givutrr 

 part. 



7. AB is the diameter of a circle, P a point on the circle, 

 PM perpendicular on AB, on A If, MB, as diameters are described 

 two circles meeting AP, BP in Q y R respectively : prove that Q R 

 touches both circles. 



8. Given two straight lines in position and a point equidistant 

 from them : prove that any circle passing through the given point 

 and the point of intersection of the given lines will intercept on 

 the given lines segments whose sum, or whose difference, is equal 

 to a given length. 



9. A triangle circumscribes a circle and from each point of 

 contact is drawn a perpendicular to the line joining the other two: 

 prove that the lines joining the feet of these perpendiculars are 

 respectively parallel to the sides of the original triangle. 



10. On a straight line AB and on the same side of it are 

 described two segments of circles, AP, AQ are chords of the two 

 segments including an angle equal to that between the straight 

 lines touching the two circles at A : prove that P, Q, B are in one 

 straight line. 



11. The centre A of a circle lies on another circle which cuts 

 the former in B, (7; AD is a chord of the latter circle meeting 

 BC in E, and from D are drawn DF, DG to touch the former 

 circle : prove that G, E, F lie on one straight line. 



12. If the opposite sides of a quadrilateral inscribed in a 

 circle be produced to meet in P, Q, and if about two of the triangles 

 so formed circles be described meeting again in R : P, R, Q will 

 lie on one straight line. 



13. Two circles intersect in A, and through A any two straight 

 lines BAC, B'AC' are drawn terminated by the two circles: provo 

 that the chords BB', CC ' of the two circles are inclined at a con- 

 stant angle. 



