156 BOOK OF MATHEMATICAL PROBLEMS. 



743. The equation of the straight line passing through the 

 centres of the inscribed and circumscribed circles is 



-- - -- - -- F , 



sin A v sm B v sm G 



Prove that the point 



q ft 



sin A (m + n cos A ) sin B (m + n cos B) sin C (m + n cos C) 

 lies on this straight line. 



744. If a, /?, y be perpendiculars from any point on three 

 straight lines which meet in a point and make with each other 

 angles A, B, C ; the equation la? + mj3* + n-f = will represent 

 two straight lines, which will be real, coincident, or imaginary 

 according as mn sin 2 A + nl sin 2 B + Im sin 2 G is negative, zero, 

 or positive. 



745. If a;, y, z be the perpendiculars from A, B, C on any 

 straight line, and a, /?, y the areal co-ordinates of any point on 

 that line, then xa. + yfl + zy = ; and the perpendicular from any 

 point (a, /?, y) on the straight line is xa + yfi + zy. 



746. "Within a triangle ABC are taken two points 0, 0' ; 

 AO, BO, CO meet the opposite sides in A', 1?, C', and the points 

 of intersection of O'A, B'C'', &B, C'A' ; O'C, A'B' are respect- 

 ively D, E, F: prove that A'D, B?E y C'F will meet in a point; 

 and this point remains the same if 0, 0' be interchanged in the 

 construction. 



747. If x, y, z be the perpendiculars from A, B t G on any 

 tangent (1) to the inscribed circle 



x sin A + y sin B + z sin G = 27? sin A sin B sin G ; 



(2) to the circumscribed circle 



x sin 2 A + y sin 2B + z sin 2(7 = 472 sin A sin B sin G ; 



(3) to the Nine Points' Circle 



x sin A cos (B-C) + y sin B cos (C - A) 



+ ztiinC cos(A-B) = 27? sin A sin B sin C j 



