584 Transactions. 



Art. LIV. — On Orthogonal Circles. 



By E. G. Hogg, M.A., F.B.A.S. 



[Read before the Philosophical Institute of Canterbury, 2nd December, 1914.'] 



The tripolar equation of the circle of radius r, which has its centre at 

 the point O, whose trilinear co-ordinates are (a (3 y ), is 



XJ=aa X + b(3 Y + c 7o Z - 2RS - 2Ar 2 = o (i) 



where S = af3y + bya + ca/3 and A is the area of the triangle of 

 reference.* 



Let d be the distance of O from H, the centre of the circle ABC. If 

 U passes through H, then 



R 2 K + &#. + c 7o ) = 2RS + 2 Ad 2 

 i.e., 2RS =2a(R 2 -<F); 



hence (i) may be written in the form 



U = aa X + &/? Y + c 7o Z = 2A(R 2 + r 2 -rZ 2 ) (ii) 



If d? = R' 2 + r 2 , the equation of the circle having its centre at (a (3 y ) and 

 cutting the circle ABC orthogonally is 



U =aa X+ b/3 Y + c 7o Z = o (iii) 



Its radius is given by the relation r 2 = — ' " S , and its trilinear 



A 

 equation is 



(ao + b/3 + Cy) j (cfo + by ) a + (ay + Ca )/3 + {ba + a^ ) y } 

 = 2 A {afiy + bya + Ca/3). 



From (iii) Ptolemy's theorem may be deduced. Let r = o so that 

 reduces to a point-circle lying on the circle ABC. Suppose its centre O to 

 lie on the minor arc BC of the circle ABC, and let D, E, F be the feet of 

 the perpendiculars from O on BC, CA, AB respectively. We then have 

 - 2R . OD = OB . OC, 2R . OE = OC . OA, 2R . OF = OA . OB. 



Substituting these for a /5 y in the equation aa X o + b(3 Y + cy Z = o, 

 we have 



AO . BO . CO [- BC . AO + CA . OB + AB . OC] = o, 



which is Ptolemy's theorem. 



From the form of the equation 



U = aa X + 6/3 Y + cy Z = o, 



it is seen that if U pass through the fixed point (XjYiZi), the locus of the 

 centre of the circles which pass through a fixed point and cut the circle 

 ABC orthogonally is the straight line whose equation is 



Xjaa + Y^/3 + ZjCy = o. 



The equation of the orthogonal circle passing through two fixed 

 points (X^ZO, (X. 2 Y 2 Z 2 ) is 



X YZ 



X 2 Y 2 Z 2 



= o. 



* Trans. N.Z. Inst., vol. 46, p. 319. 



