Elementary Modifications of Rogers' and Alyar's Theorems,

Takashi byHAMADA, Urawa.

Several geometricians have generalized Feuerbach's theorem, since it was found out more than one hundred years ago. I have also proved its simple extension(1.). Theorem I. A triangle A1A2A3and a point P are given in a plane. Let L1,L2; L3 be the middle points of the sides of A1A2A3and let PM1,PM2, PM3 be the perpendiculars on them. Let a circle S cut the three circles Wthogonally, whose centres are L1, L2, .L3. and their radii L1M1, L2M2, .,Gelds respectively. Then, the circle S touches the nine-point circle of A1A2A3. Moreover, if the radius of S and the distance between , O (the circum-centre of A1A2A3) and P .be denoted by r and ƒÂ respectively, then we have `'

Hereby R denotes the• radius of. the circumcircle of A1A2A3.It: is evident that the former part of the theorem is. an extension of Feuerbach's theorem. The latter part is regarded as an extention of Chappel's rela- tion. I saw that this theorem (the former part) is, however, no other than an elementary modification of Rogers' extension of Feuerbach's theorem. And Rogers' theorem is proved geometr ically by the application of Aiyar,s theorem, so we may treat the former as a special case of the latter. Therefore, it appears that Aiyars theorem is also to be men- tioned elementarily, and it may be stated as follows. Theorem, II. A triangle A1A2A3and a line x are given in a,,plane. From any two points P, Q .on x draw perpendiculars PM1,PM2,

(1) Tokyo BatsurigakkO Zaashi, No. 561. MODIFCATIONSOFROE ERfS' AND AIYAWS THEOREMS. PM3 and QN1,QN2,QN3 on the sides of A1A2A3respectively. There exists a circle S which cuts three ' eircles, whose centres are N1,N2,N3 and their radii respectively N1M1,N2M2, N3M3,ortho- gonally. Then, Scuts the ortho-polar circle of x with respect to A1A2A3 orthogonally. I was not able to prove Theorem II purely geometrically. But, for. the proof, the following matter is. only. to be shown. The line x cuts the circle Sin Y, Z. From Y and Z draw per-. pendiculars y. and z respectively. Then the 'conic that touches the sides of A1A2A3 and y, z should have the same centre as S, for S must be the joint-director circle of this conic and its confocal conic touching x. I proved the theorem by means of orthogonal coordinates.

Proof of Theorem .II. Let the line x be x-axis of orthogonal coordinates and let PQ=m, Q (L,0).(i=1,2,3). Let' ƒ¿ be the angle at which QN1 in- tersects x-axis. hen,T M1M1 is equal to msinƒ¿1, and the equations of A1Ak and

QN1, are, A1Ak: (1).

(2) The coordinates of N1 are derived from (1) (2).

(3) The circle S, cuts these three circles (3) orthogonally.

(4) (The second row in the determinant in (4) represents rows from 118 TAKASHI HAMADA

Fig. 1.

Fig. 6.

the second to the fourth). The determinant in (4) is divided into two parts as follows: MODIFICATIONSOF , ROGERS' AND OIYAR'S THEOREMS. 117

(5) Let l 'be fixed and m change, that is, let. Q fix and P move on x Then, S forms a coaxalsystem. This,follows immediately from the form of (5). If we make m infinitely great, then we have the egtia tion of the radical axis of the, system, andit is what the first term in (5) equated to zero:

(6) The determinant in (6) is divided into two as follows :

(7)

The first term in (7) is equal to zero, for its first two columns are equal. The `second term is divided-into two as follows:,

(8) (8) is the equation of the radical axis of the system obtained from (6). Finally, let l vary or let Q vary on x, then (8) forms a pencil of rays. The centre of this pencil is the orthopole T of x with respect to A1A2A3.For, if Q coincides with one of the poiits Q1,02 of intersection of x and the circumcircle of A1A2A3,the coaxal systems correspondingto Q1,Q2,have the Simson lines of Qi,Q2 with respect to, A1A2A3as their radical axes respectively. But it is known that such Simson lines intersect at the ortho- pole T of x with respect to A1A2A3.Therefore, the centre of the pencil is the orthopole T of x Moreover,the pedal, circles of all Q, as it is eapiiy.seen, belong to these systems. And these pedal circles cut the ortho-polar circle of x with respect to A1A2A3ortho- g6nally (Lemoyne'stheorem). Thus, it was proved that all circles which belongto the systems cut the ortho-polar circle of x orthogonally. Q.E.D. 118 HAMADA: ROGERS' AND AIYAR'S THEOREMS.

It is evident that -this theorem coincides with Aiyar's, since there are •‡2 cirolep or •‡1 coaxal system generated by P, Q on x. That S forms a coaxal system when P moves and Q is fixed, is shown elementarily as follows:. Let P, P be two positions of P and let S, S1 be two circles cor- responding to P, P1 respectively. Let t1 t2 denote the lengths of tangents drawn to S, 81. from N1. Then we have

hence, Therefore, N1 lies on a circle coaxal. with S and S, or the pedal circle N1N2N3of Q is coaxal with S and S'. If we take another point P" on x, we get a circle S" coaxal with N1N2N3and S. Therefore, it is shown that S, S', S" and the pedal circle N1N2N3 form a coaxal system. Q.E.D. What is called as Lemoyne's theorem, is the fact that the pedal circle N1N2N3of a point Q on x belongs to the system. If Q coincides with the circumcentre of A1A2A3,the orthopolar circle S of x reduces to a point T on the nine-point circle of A1A2A3. In this case the coaxal system generated by the variation of P (on OP) is parabolic, and it contains the nine-point circle. Thus Theorem I is obtained by the help of Theorem II. For reference, the original forms of Rogers' and Aiyar's their rems are the following. L. J. Rogers' Theorem. If a conic touches three sides of a triangle, its joint director circle with either confocal drawp through the circumcentre touches the nine-point circle Aiyar's Theorem. The ortho-polar circle of a line x, which touches a conic confocal with another conic touching. the three sides of a given triangle, with respect to the triangle cuts the joint-director circle of these two conics orthogonally.

(Receivedthe 31th,January, 1942.)