168 BOOK OF MATHEMATICAL PROBLEMS. 



805. A triangle is self-conjugate to the conic 



fo'-f 7?i 2 + wy f = 0, 

 and two of its sides touch the conic 



prove that the envelope of the third side is the conic 

 la 9 



MMH 



I 



m n^ 

 and that this will coincide with the second if 



I m n 



T , +,+ , = 0. 



I m n 



Prove also that the three conies have four common tangents, and 

 that the locus of the point of intersection of the two sides is 

 the conic 



fm ri\ , 

 ( + -)a 3 

 \m nj 



IX. Anharmonic Properties. 



The anharmonic ratio of four points A, JB, C, D in one straight 



, ^ , AB AC AB . CD 



line, denoted by [ABCD], means the ratio jjj : CD' r AC~~BD ' 



the order of the letters marking the direction of measurement of 

 any segment, and segments measured in opposite directions 

 being affected with opposite signs. So, if A, , C, D be any 

 four points in a plane, and P any other point in the same plane, 



z> f A /*m i 8 ^ n APB . siri CPU ,, 



P {ABCD} denotes - -p^ . ~ , the same rules being ob- 

 served as to direction of measurement and sign for the angles in 

 this expression as for the segments in the former. 



Either of these ratios is said to be harmonic when its 

 value is 1; in which case AD is the harmonic mean between 

 AB and AC, and DA the harmonic mean between DB and DC. 



