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
tranquanghung@hus.edu.vn
Nguyen Thi Thuy Duong
Concord College, Shrewsbury, Shropshire, United Kingdom
nt.thuyduong3402@gmail.com
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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