Hogg. — Theorems relating to Sub-Polar Triangles. 327 



Art. XLI. — Some. Theorems relating to Sub-Polar Triangles. 



By Evelyn G. Hogg, M.A., Christ's College, Christchurch. 



{Read before the Philosophical Institute of Canterbury, 1st December, 1909.] 



§1. Let the straight lines joining the vertices of the triangle ABC to the 

 points Oi, O, meet the opposite sides BC, CA. AB, in DjD,, EiE,, FiF.„ 

 respectively ; then the triangles DiEjFi, D3E2F2 are termed sub-polar 

 triangles, the points Oj, O.2 bemg their respective poles. 



The vertices of any two sub-polar triangles lie on a conic, for, taking 

 the co-ordinates of the points Oj, O.2 to be (aiySiyj), (aa/^aya) respectively, 

 it may be at once verified that the conic 



a2 B'- 72 



\^i y-i ft 71/ \7i a-2. 7i °-i/ 



- a/3 ( '. -f -V) = (i) 



passes through the vertices of DiEjFi and D2E2F2. 



If the points Oi. O2 be regarded as being determined by the inter- 

 section of the line L ez /a + mfi + ny = with the conic S ^ ao/^'y 

 + ySoya + y„ a/3 = 0, then the conic (i) takes the form 



la" viB'^ ny- By , „ 1 \ "Y"- , - ,-> \ 



- + -— + — + - — [ml3o + njo - la„) + ('iyo + ia,, ~ m^J 



0-0 Po 7o Pu7o 7oao 



-f -~ {la, + m/3o - nyo) = . . ... . (ii). 



CloPo 



Let the line L = pass through the fixed point 0'(a'/3'y') ; then, 

 eliminating I between equation (ii) and the relation la -)- w/3' + wy' = 0, 

 the conic (i) takes the form wiSi + "S.2 = 0, when 



Q _ , «^ , 72 , ' 1 ' \ ^ /'>' "^ y« / .• <\ 



^^ == ^ «; - " % + (^"'^ + ^"-^^oU ~ a;j - y;ao^y''o - yo«) - o. 



Hence we derive the theorem, — 



If the poles of two sub-j)olar triangles be determined by the iidersection 

 of a variable line passing through a fixed point tvith a fixed conic circum- 

 scribing the triangle of reference, the conic circumscribing the two sub-polar 

 triangles passes through fotir fixed points. 



It will be seen on inspection that one of the points of intersection of 

 the two conies S', S" is the point (ao^ojo) — ^he pole of the conic So. 



A particular case arises if we suppose the variable line L to pass 

 through the point (ao/3oyo)- Conic (ii) reduces to 



— + ~i^ + — - 2/ao f^ - 2m^^ — 2wyo -^ = . . (ill) 



subject to the relation la., + m/3,j + uy, = o. By eliminating I we obtain 



-/?o(^ - ^) {:"- + 1 ^ -') - -To(-^- - -) ( " + f + ■-^) = «. 



Va,, /3o/ Vao Po 7o/ ^70 ao/ \a" ^0 7u/ 



which shows that all conies of this family pass through the four fixed 

 points (ao^oyo), ( - ^aof^^yo), ("o - 3/3oyo), {a^k, - 3yo). 



