GEOMKTKV. 7 



of the otli in a poiiit: prove that straight lines drawn at 



1 >ondiiig angular points of the second parallel t 

 of the first will meet in a point; and that each triangle will In- 

 divided into three triangles, which are each to each in the same 

 ratio. 



44. In any triangle ABC, 0, & are the centres of the 

 U-cl ciivle, and of the escribed circle opposite A ; 00' meets 



BO in D y any straight line through D meets AB t AC respectively 

 in b y c, Ob, O'c intersect in P, Oc, (76 in Q : prove that P, A, Q 

 lie on one straight line perpendicular to 00'. 



45. The centre of the circumscribed circle of a triangle, and 

 the centre of perpendiculars are joined: pnvr that the j> 

 line is divided into segments in the ratio of 1 : 2 by each of the 



ht lines joining the angular points to the middle points of 

 the o.oMte sides. 



46. Inscribe a parallelogram in a given triangle so that its 

 uals may intersect at a given point within the triangle. 



The side BC of a triangle ABC is bisected in />, a 

 straight line parallel to BC meeting AB t AC prodiuvd in /', /' 

 respectively is divided in Q so that PQ t BD, QP* are in continued 



rough Q is drawn a straight line RQK t- 



nated by AB, AC and bisected in Q : prove that the triangles 

 ABC, ARK are equal. 



48. On Aft, AC two sides of a triangle are taken two p* 



/'. / ; J /;. .!< r. produced to F, 6 so that BF is equal to AD 



fit to .1 /. ; / .ed the two former ni 



in // hat the triangl it equal to the two triangles 



/;//<, .!/>/: together. 



49. If two sides of a triangle be given in position :ml their 

 1* also given, and if the third aide be divided in a given 



li vision will He on one of two fixed straight 



