GEOMKTKY. 3 



14. Four points are taken on a circle and the three pairs of 



_iit lint > \\liich can be drawn through the four points inti-r- 



sect respectively in E, F, G : prove that the three pairs of straight 



lines which bisect the angles at E, F, G respectively will be in the 



same directions. 



1 -~>. Through a point of intersection of two circles is drawn a 

 straight line at right angles to their common chord and terminated 

 by the circles, and through the other point is drawn a straight line 

 equally inclined to the straight lines joining that point to the 

 extremities of the former straight line: prove that the tangents to 

 the two circles at the points on this latter straight line will inter- 

 sect in a point on the common chord. 



16. Two circles cut each other at A, and a straight line SAC 

 is drawn terminated by the circles; with B, C as centres are 

 described two circles each cutting at right angles one of tl. 

 circles : prove that these two circles and the circles of which BC 

 is a diameter will have a common chord. 



1 7. Circles are described on two of the sides of a triang 1 

 ters, and each meets the perpendicular from the opj 



angular point on its diameter in two pointer prove that these 

 points lie on a circle. 



18. The tangents from a point to a circle are bisected by a 

 straight line which meets a 'J of the circle in R : \ 

 that the angles HOP, OQR are equal 



A straight line PQ of given leu tercepted between 



two straight lines OP, OQ %\\>\\ in position; through /*, Q are 

 drawn straight lines in given din >terseoting iu 



and the angles POQ, PRQ arc equal and on the tame side of 

 prove that R lies on a fixed n 



From the point of intersection of the diagonals of a 

 nil inscribed in a circle perp- 



; incite angles < 

 i*e perpendicular* 

 of one of the angles between the diagonals of the former. 



