For any triangle, nine significant points, including the midpoints of its sides, the feet of its altitudes, and the midpoints of the segments from each vertex to the orthocenter, all lie on a single circle. A classical result of triangle geometry, sometimes attributed jointly to several nineteenth-century geometers.
Facts
StatementFor any triangle, a single circle passes through nine significant points: the midpoint of each side, the foot of each altitude, and the midpoint of the segment from each vertex to the orthocenter. 1 Classification
Statement Form Connections
Has Statement Form
In Branch
Principal Result Of
Source Olry Terquem (Wikipedia)
Proved By
Source ProofWiki: Nine-Point Circle Theorem - ProofWiki
Sources
1. Nine-Point Circle Theorem (Wikipedia)
Wikimedia Foundationlead paragraph, theorem statement sentenceQuote, lead paragraph, theorem statement sentence
In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle.
2. ProofWiki: Nine-Point Circle Theorem - ProofWiki
ProofWikiHistorical noteQuote, Historical note
The Nine Point Circle Theorem was proved in $1820$ by Jean-Victor Poncelet and Charles Julien Brianchon.
Olry Terquem (Wikipedia)
Principal Result Of: Olry Terquem, Lead, principal resultQuote, Principal Result Of: Olry Terquem, Lead, principal result
In geometry, Terquem is known for naming the nine-point circle and fully proving its properties.
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