Hogg. — On certain TrIpoJar lielations. 349 



j 4. If Ji, d.2, d.^ be the lengths of the sides of the pedal triangle of the 

 point P (a,^iyi), then 



di^= X sin-^A, di = Y sin^B, d^ = Z sin^C. 



Hence if the sum of the squares of the sides oi the pedal triangle of P 

 be given, the locus of P is the circle a^X + 6^Y + c^Z = k, a circle whose 

 centre is at the symmedian point of the triangle ABC. 



The minimum value of d^^^ d.,^ + d.? is -— ,- . 



2 (a2) 



The locus of points whose pedal triangles are right-angled is 



a^X = h^Y + c^Z 

 and two similar circles. 



The above circle passes through B and C and has its centre at the 

 point in which the tangents to the circle ABC at B and C intersect. 



If the pedal triangle of P be equilateral, then a^x = ^^Y = c^Z = k. 

 Substituting in relation (i) we obtain the quadratic equation in k 



k2 [2 (a*) - 2 (Z'^c^)] - Ka^b^c'' 2 (a^) + a^h^c" = o. 



The roots of this equation are real, showing that every triangle has two 

 points whose pedal triangles are equilateral. 



It is easily deduced that the areas of the two triangles are 



V3A'' [2K)+V3a1 ^^^^ V3 A^ [2 (g^; - V3 a] 

 2 [2(a*) - 2(62c2)] 2 [2 (a*) - 2 (^^c^)] 



The coaxial circles a^X = b^Y — c^Z are the circles of Apollonius, 

 and their common points P, P' lie on the Brocard diameter. 



The distance between P and P' is - ,— ; ^ , „ „ 



2 (a*) - 2(62c2) 



i 5. If P (ai^iYi) and Q l- r ^) ^^ isogonal conjugates, and if 



AP = Pi, BP = P,, CP = P„ AQ = Qi, BQ = Q.3, CQ = Q3. then 

 Pi2 sin^A = /?i2 + ^^2 ^ o^^^j COS A 



Qx^sin^^ = K-^(l +i, + ^^) 

 , P,2 sm^A 



hence Qi^yi = + k^P,, Q.^y^aj = ± k-^P,, q,a,l3, = +K»P,. 

 Multiplying these in turn by aai, b/3i, Cyi and adding 



«A7i {aQ^ + hQ, + cQ,} = k2 (aPjai + bF^ ± cP.y^) 



Also K^(-+ ^ + '-] = 2a, 



hence (-T^iSiyi + ^yiai + Ca^Bi) (aQi + ^Q.^ + cQs) 



= 2 A (aPiai + 6Po^, ± cP.,y,). 

 If d be the distance between (ai/3iyi) and the circum-centre, then 



Hence 



9R 



R' — «' 



