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Forum Geometricorum Volume 5 (2005) 17–20.

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FORUM GEOM ISSN 1534-1178

A Synthetic Proof of Goormaghtigh’s Generalization of Musselman’s Theorem Khoa Lu Nguyen

Abstract. We give a synthetic proof of a generalization by R. Goormaghtigh of a theorem of J. H. Musselman.

Consider a triangle ABC with circumcenter O and orthocenter H. Denote by B ∗ , C ∗ respectively the reflections of A, B, C in the side BC, CA, AB. The following interesting theorem was due to J. R. Musselman. A∗ ,

Theorem 1 (Musselman [2]). The circles AOA∗ , BOB ∗ , COC ∗ meet in a point which is the inverse in the circumcircle of the isogonal conjugate point of the nine point center. B∗

A C∗

Q

N∗ H

O N

B

C

A∗

Figure 1

R. Goormaghtigh, in his solution using complex coordinates, gave the following generalization. Theorem 2 (Goormaghtigh [2]). Let A1 , B1 , C1 be points on OA, OB, OC such that OB1 OC1 OA1 = = = t. OA OB OC (1) The intersections of the perpendiculars to OA at A1 , OB at B1 , and OC at C1 with the respective sidelines BC, CA, AB are collinear on a line . (2) If M is the orthogonal projection of O on , M the point on OM such that OM  : OM = 1 : t, then the inversive image of M in the circumcircle of ABC Publication Date: January 24, 2005. Communicating Editor: Paul Yiu. The author thanks the communicating editor for his help and also appreciates the great support of his teacher Mr. Timothy Do.

18

K. L. Nguyen

is the isogonal conjugate of the point P on the Euler line dividing OH in the ratio OP : P H = 1 : 2t. See Figure 1.

A

A1 O B1 H

P

B

X

C1 P∗

C Z

M

Y

Figure 2

Musselman’s Theorem is the case when t = 12 . Since the centers of the circles OAA∗ , OBB ∗ , OCC ∗ are collinear, the three circles have a second common point which is the reflection of O in the line of centers. This is the inversive image of the isogonal conjugate of the nine-point center, the midpoint of OH. By Desargues’ theorem [1, pp.230–231], statement (1) above is equivalent to the perspectivity of ABC and the triangle bounded by the three perpendiculars in question. We prove this as an immediate corollary of Theorem 3 below. In fact, Goormaghtigh [2] remarked that (1) was well known, and was given in J. Neuberg’s M´emoir sur le T´etra`edre, 1884, where it was also shown that the envelope of  is the inscribed parabola with the Euler line as directrix (Kiepert parabola). He has, however, inadvertently omitted “the isogonal conjugate of ” in statement (2). Theorem 3. Let A B  C  be the tangential triangle of ABC. Consider points X, Y , Z dividing OA , OB  , OC  respectively in the ratio OY OZ OX = = = t. (†)   OA OB OC  The lines AX, BY , CZ are concurrent at the isogonal conjugate of the point P on the Euler line dividing OH in the ratio OP : P H = 1 : 2t. Proof. Let the isogonal line of AX (with respect to angle A) intersect OA at X  . The triangles OAX and OX A are similar. It follows that OX · OX = OA2 , and X, X  are inverse in the circumcircle. Note also that A and M are inverse in the

Goormaghtigh’s generalization of Musselman’s theorem

19

A

P H

O

M

B

X

C

X

A

Figure 3

same circumcircle, and OM · OA = OA2 . If the isogonal line of AX intersects the Euler line OH at P , then OX  OX  1 OA 1 OP = = = · = . PH AH 2 · OM 2 OX 2t The same reasoning shows that the isogonal lines of BY and CZ intersect the Euler line at the same point P . From this, we conclude that the lines AX, BY , CZ intersect at the isogonal conjugate of P .  For t = 12 , X, Y , Z are the circumcenters of the triangles OBC, OCA, OAB respectively. The lines AX, BY , CZ intersect at the isogonal conjugate of the midpoint of OH, which is clearly the nine-point center. This is Kosnita’s Theorem (see [3]). Proof of Theorem 2. Since the triangle XY Z bounded by the perpendiculars at A1 , B1 , C1 is homothetic to the tangential triangle at O, with factor t. Its vertices X, Y , Z are on the lines OA , OB  , OC  respectively and satisfy (†). By Theorem 3, the lines AX, BY , CZ intersect at the isogonal conjugate of P dividing OH in the ratio OP : HP = 1 : 2t. Statement (1) follows from Desargues’ theorem. Denote by X  the intersection of BC and Y Z, Y  that of CA and ZX, and Z that of AB and XY . The points X , Y  , Z  lie on a line . Consider the inversion Ψ with center O and constant t · R2 , where R is the circumradius of triangle ABC. The image of M under Ψ is the same as the inverse of M  (defined in statement (2)) in the circumcircle. The inversion Ψ clearly maps A, B, C into A1 , B1 , C1 respectively. Let A2 , B2 , C2 be the midpoints of BC, CA, AB respectively. Since the angles BB1 X and BA2 X are both right angles, the points B, B1 , A2 , X are concyclic, and OA2 · OX = OB · OB1 = t · R2 .

20

K. L. Nguyen Y

A

A1 Z

C2

B2

A3 O

B1

C1 A2

X

B

C X

Figure 4

Similarly, OB2 · OB2 = OC2 · OC2 = t · R2 . It follows that the inversion Ψ maps X, Y , Z into A2 , B2 , C2 respectively. Therefore, the image of X under Ψ is the second common point A3 of the circles OB1 C1 and OB2 C2 . Likewise, the images of Y  and Z  are respectively the second common points B3 of the circles OC1 A1 and OC2 A2 , and C3 of OA1 B1 and OA2 B2 . Since X  , Y  , Z  are collinear on , the points O, A3 , B3 , C3 are concyclic on a circle C. Under Ψ, the image of the line AX is the circle OA1 A2 , which has diameter OX  and contains M , the projection of O on . Likewise, the images of BY and CZ are the circles with diameters OY  and OZ  respectively, and they both contain the same point M . It follows that the common point of the lines AX, BY , CZ is the image of M under Ψ, which is the intersection of the line OM and C. This is the antipode of O on C. References [1] R. A. Johnson, Advanced Euclidean Geometry, 1925, Dover reprint. [2] J. R. Musselman and R. Goormaghtigh, Advanced Problem 3928, Amer. Math. Monthly, 46 (1939) 601; solution, 48 (1941) 281 – 283. [3] D. Grinberg, On the Kosnita point and the reflection triangle, Forum Geom., 3 (2003) 105–111. Khoa Lu Nguyen: 306 Arrowdale Dr, Houston, Texas, 77037-3005, USA E-mail address: [email protected]

A Synthetic Proof of Goormaghtigh's Generalization of ...

Jan 24, 2005 - C∗ respectively the reflections of A, B, C in the side BC, CA, AB. The following interesting theorem was due to J. R. Musselman. Theorem 1 (Musselman [2]). The circles AOA. ∗. , BOB. ∗. , COC. ∗ meet in a point which is the inverse in the circumcircle of the isogonal conjugate point of the nine point center.

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