ICO BOOK OF MATHEMATICAL PROBLEMS. 



7G3. A straight lino drawn through the centre of the in- 

 scribed circle meets the sides of the triangle ABC in A', B, <?', 

 these points are joined to the centres of the corresponding escribed 

 circles; prove that the joining lines meet two and two on the 

 sides of the triangle ; and, if A", B", C" be their points of inter- 

 section, the circles on A' A", B'B", C'C" as diameters will touch 

 each other in one point lying on the circumscribed circle ; and 

 their common tangent will be normal to the circumscribed circle. 



7G4. A circle meets the sides of a triangle ABC in P, P' ; 

 Q, Q ; R, R f respectively, and AP, BQ, CR meet in the point 

 la = m/3 ny j prove that AP' t BQ f ) CR meet in the point 

 Fa = m' = n' where 



7C5. The equation of the circle which passes through the 

 centres of the escribed circles of the triangle of reference is 



bca' + cap 2 + db-f + (a + b + c) (afiy + lya + cafi) = 0. 



7C6. The lines joining the feet of the perpendiculars meet 

 the corresponding sides of the triangle ABC in A', B', C'; prove 

 that the circles on AA\ BB', CC' as diameters will touch each 

 other if 



sec* A + sec* B + sec 2 C + 2 sec A sec B sec C = 7. 



7G7. If the side BC subtend angles 0, & at the foci of an 

 inscribed conic, 



sin (6 - A) sin (ff-A) = sin sin &. 



768. If P be any point on the minimum ellipse circum- 

 scribing a given triangle ABC, 



AP BP CP 



bin B1 J G + sin (JPA + ETZPi ~ ' 



cot BPC + cot CPA + cot APB = cot A + cot B + cot C ; 



the angles JBPC, CPA y APB being so taken that their sum 

 is 360. 



