Editing Orthocenter (section)

==History==
The theorem that the three altitudes of a triangle concur (at the orthocenter) is not directly stated in surviving [[Greek mathematics|Greek mathematical]] texts, but is used in the ''[[Book of Lemmas]]'' (proposition 5), attributed to [[Archimedes]] (3rd century BC), citing the "commentary to the treatise about right-angled triangles", a work which does not survive. It was also mentioned by [[Pappus of Alexandria|Pappus]] (''Mathematical Collection'', VII, 62; {{c.}} 340).<ref name=newton>
{{cite book |last=Newton |first=Isaac |author-link=Isaac Newton |editor-last=Whiteside |editor-first=Derek Thomas |year=1971 |title=The Mathematical Papers of Isaac Newton |volume=4 |publisher=Cambridge University Press |chapter=3.1 The 'Geometry of Curved Lines' |pages=454–455 |chapter-url=https://archive.org/details/mathematicalpape0004newt/page/454/ |chapter-url-access=limited }} Note Whiteside's footnotes 90–92, pp. 454–456.
</ref> The theorem was stated and proved explicitly by [[Alī ibn Ahmad al-Nasawī|al-Nasawi]] in his (11th century) commentary on the ''Book of Lemmas'', and attributed to [[Abu Sahl al-Quhi|al-Quhi]] ({{floruit|10th century}}).<ref>{{cite journal |last1=Hajja |first1=Mowaffaq |last2=Martini |first2=Horst |year=2013 |title=Concurrency of the Altitudes of a Triangle |journal=Mathematische Semesterberichte |volume=60 |number=2 |pages=249–260 |url=https://www.researchgate.net/publication/257442911 |doi=10.1007/s00591-013-0123-z
}}<br>
{{cite journal |last=Hogendijk |first=Jan P. |title=Two beautiful geometrical theorems by Abū Sahl Kūhī in a 17th century Dutch translation |journal=Tārīk͟h-e ʾElm: Iranian Journal for the History of Science |volume=6 |pages=1–36 |year=2008 |url=https://www.sid.ir/paper/146602/en }}
</ref>

This proof in Arabic was translated as part of the (early 17th century) Latin editions of the ''Book of Lemmas'', but was not widely known in Europe, and the theorem was therefore proven several more times in the 17th–19th century. [[Samuel Marolois]] proved it in his ''Geometrie'' (1619), and [[Isaac Newton]] proved it in an unfinished treatise ''Geometry of Curved Lines'' {{nobr|({{c.}} 1680).<ref name=newton />}} Later [[William Chapple (surveyor)|William Chapple]] proved it in 1749.<ref>{{cite journal |last=Davies |first=Thomas Stephens |author-link=Thomas Stephens Davies |year=1850 |title=XXIV. Geometry and geometers |journal=[[Philosophical Magazine]] |series=3 |volume=37 |number=249 |pages=198–212 |url=https://zenodo.org/record/1919807 |doi=10.1080/14786445008646583}} [https://archive.org/details/londonedinburg3371850lond/page/207 Footnote on pp. 207–208]. Quoted by {{cite web |first=Alexander |last=Bogomolny |author-link=Alexander Bogomolny |year=2010 |url=https://www.cut-the-knot.org/triangle/Chapple.shtml |title=A Possibly First Proof of the Concurrence of Altitudes |work=Cut The Knot |access-date=2019-11-17}}</ref>

A particularly elegant proof is due to [[François-Joseph Servois]] (1804) and independently [[Carl Friedrich Gauss]] (1810): Draw a line parallel to each side of the triangle through the opposite point, and form a new triangle from the intersections of these three lines. Then the original triangle is the [[medial triangle]] of the new triangle, and the altitudes of the original triangle are the [[perpendicular bisector]]s of the new triangle, and therefore concur (at the circumcenter of the new triangle).<ref>
{{cite book |last=Servois |first=Francois-Joseph |author-link=Francois-Joseph Servois |title=Solutions peu connues de différens problèmes de Géométrie-pratique |language=fr |trans-title=Little-known solutions of various Geometry practice problems |publisher=Devilly, Metz et Courcier |year=1804 |page=15
}}<br>
{{cite book |contributor-last=Gauss |contributor-first=Carl Friedrich |year=1810 |contributor-link=Carl Friedrich Gauss |last=Carnot |first=Lazare |translator-last=Schumacher |title=Geometrie der Stellung |contribution=Zusätze |language=de }} republished in {{cite book |last=Gauss |first=Carl Friedrich |title=Werke |volume=4 |chapter-url=https://archive.org/details/werkecarlf04gausrich/page/n405/ |chapter=Zusätze |page=396 |publisher= Göttingen Academy of Sciences |year=1873
}}<br>
See {{cite journal |last=Mackay |first=John Sturgeon |author-link=John Sturgeon Mackay |year=1883 |title=The Triangle and its Six Scribed Circles §5. Orthocentre |journal=Proceedings of the Edinburgh Mathematical Society |volume=1 |pages=60–96 |doi=10.1017/S0013091500036762 |doi-access=free}}</ref>