Journal for Geometry and Graphics 25 (2021), No. 1, 061--069
Copyright Heldermann Verlag 2021
Generalizations of Fagnano's Problem
Tran Quang Hung
High School for Gifted Students, Vietnam National University at Hanoi, Hanoi, Vietnam
Nguyen Thi Thuy Duong
Concord College, Shrewsbury, Shropshire, United Kingdom
We generalize Fagnano’s famous problem of minimal inscribed perimeter by replacing the orthocenter with an arbitrary interior point P. By adding weights associated with P to Fagnano’s inequality, we show that the new, generalized expression reaches minimum for the pedal triangle of P. We then further generalize our main theorem and derive some extensions by relating them to Fermat-Torricelli problem.
Keywords: Fagnano's inequality, generalized theorem, extremum problem.
MSC: 51M04, 51M16
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