Two Pedal Circles Through a Common Midpoint

If the midpoint of two points belongs to both of their pedal circles with respect to a triangle, it belongs to the nine-point circle. A proof with directed angles allows feet on side extensions. Exact examples and counterexamples show the roles of the two circle conditions and of the midpoint requirement.

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-11-02
DOI
https://doi.org/10.5281/zenodo.23887914
Primary Topic
Mathematics and Applications
Type
preprint
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preprint

Two Pedal Circles Through a Common Midpoint

Ziping Guo
1 links
Zenodo (CERN European Organization for Nuclear Research)
Mathematics and Applications
preprint

Two Pedal Circles Through a Common Midpoint

Ziping Guo
preprint en

Abstract

If the midpoint of two points belongs to both of their pedal circles with respect to a triangle, it belongs to the nine-point circle. A proof with directed angles allows feet on side extensions. Exact examples and counterexamples show the roles of the two circle conditions and of the midpoint requirement.

Zenodo (CERN European Organization for Nuclear Research)
Mathematics and Applications
DOIOpen Access PDFOpenAlex
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Two Pedal Circles Through a Common Midpoint — Ziping Guo · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS