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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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