065: on a Theorem of Brianchon and Poncelet

In a note of Margetson and Buckingham 4, they treat an interesting result of planar geometry. After we read it, we noticed that the following extension of their result holds. Theorem 1. If the vertices of 4ABC lie on a given perpendicular hyperbola, then the orthocenter H also lies on the hyperbola, and the nine-point circle passes through the symmetric center of the hyperbola. According to 3, or 11, the nine-point circle was rst found by Brianchon and Poncelet 22. The circle passes through the nine important points of 4ABC; Unfortunately, w e found that Theorem 1 is not new. According to 3, it was already proved by Brianchon and Poncelet 2. However, we could not obtain it. In the note, we consider a converse problem of the following type. Theorem 2. I f a p oint lies on the nine-point circle of a given 4ABC, where 6 = D;E;F, then there i s a p erpendicular hyperbola which has as the symmetric center and passes through the vertices A; B; C. To prove it, we use the following lemma. Lemma. Two points A and B of the xy-plane lie on the same perpendicular hyperbola y = k=xif and only if the line AB meets both x-and y-axis at points, We think that the above lemma alone is suuciently interesting. In the rest of the note, we give the proofs of Theorems 1 and 2. The proof of Theorem 1 is similar to that of 4 but quite diierent from that of 3. Proof of Theorem 1. We denote by y = k=xthe perpendicular hyperbola, and by ~ a = a; k=a, ~ b = b; k=b, ~ c = c; k=c the position vectors of A; B; C, respectively. We consider a point H 0 whose position vector ~ h = h; k=h, where h = ,k 2 =abc. We shall prove that H 0 coincides with the orthocenter H. Since we h a ve abch = ,k 2 , w e obtain that ~ a ~ h + ~ b ~ c = ah + bc 1 + k 2 abch = 0 : 1 Similarly, w e obtain that ~ a ~ h + ~ b ~ c = ~ b ~ h + ~ c ~ a = ~ c ~ h + ~ a ~ b = 0 : 2 By using 2, we obtain …