A synthetic proof of Goormaghtigh’s generalization of Musselman’s theorem. (English) Zbl 1078.51507
Let \(ABC\) be a triangle with circumcenter \(O\) and orthocenter \(H\) and let \(A^*\), \(B^*\), \(C^*\) be the reflection of \(A\), \(B\), \(C\) in the side \(BC\), \(CA\), \(AB\) respectivily. Then the circles \(AOA^*\), \(BOB^*\) and \(COC^*\) meet in a point which is the inverse in the circumcircle of the isogonal conjugate point of the nine point center.
[Concerning the theorem of Musselman see J. R. Musselman and R. Goormaghtigh, “Advanced Problem 3928”, Am. Math. Mon. 46, 601 (1939) and “Solution to Advanced Problem 3928”, Am. Math. Mon. 48, 281–283 (1941).] The author gives a synthetic proof of an extensive generalization of this theorem (see the same references).
[Concerning the theorem of Musselman see J. R. Musselman and R. Goormaghtigh, “Advanced Problem 3928”, Am. Math. Mon. 46, 601 (1939) and “Solution to Advanced Problem 3928”, Am. Math. Mon. 48, 281–283 (1941).] The author gives a synthetic proof of an extensive generalization of this theorem (see the same references).
Reviewer: Erhard Quaisser (Potsdam)
