r.EOMr.Tiiv. 5 



If /> l.e the middle point of the side EC of a triangle 

 ABC and the tangents at B, C to the circuin> -rib. -d circle meet in 

 .1', the angles BAA', CAD will be equal. 



. The aide BC of ;i triangle ABC is bisected in Z>, and 

 on DA is taken a P such that the rectangle DP, DA is 



equal to the rectangle DB t DC: prove that the angles BPC, BAG 



Aether equal to two right angles. 



30. If the circle inscribed in a triangle ABC touch BC in D, 

 the circles inscribed in the triangles ABD, ACD will touch each 

 other. 



31. Given the base and the vertical angle of a triangle, prove 

 that the centres of the four circles which touch the sides : 



_:le, will lie on two fixed circles passing through the extremi- 

 ties of the base. 



A circle is drawn through B, C and the centre of perpen- 

 diculars of a triangle ABC, D is the middle point of BC and .1 !> 

 is produced to meet the circle in K : prove that AJS is bisected 



no straight lines joining the centres of the four circles 

 li the sides of a triangle are bisected by th< 

 sorib. also the middle point of the line joining any two 



>f the centres and that of tho lines joining the other two are the 

 extremities of a diameter of the circumscribed circle. 



With three given points not lying in one straight 

 AS centres, describe three circles which shall hare three common 



iits. 



: om the angular points of a triangle straight lines are 

 drawn ]xrpendicular to the opposite sides and terminated by the 



ascribed circle: prove that the parts of these lines i- 

 oepted between their point of intersection and the circle SIM 



by tl,- r.,rn-.,p..i,ding sid,-s 



