Carnot's Theorem relates the signed distances from a triangle's circumcenter to each of its three sides, stating that their sum equals the sum of the triangle's circumradius and its inradius, with a sign convention that turns negative for a side beyond which the circumcenter lies. Named for Lazare Carnot, it is a classical result of triangle geometry connecting a triangle's two principal associated circles.
Facts
StatementFor any triangle ABC, the signed sum of the perpendicular distances from the circumcenter to the three sides equals the sum of the circumradius and the inradius. 1 Classification
Statement Form Connections
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In Branch
Source Carnot's theorem (inradius, circumradius) (Wikipedia)
Proved By
Source Carnot's Theorem (Wikipedia)
Sources
1. Carnot's Theorem - MathWorld
Main entryQuote, Main entry
Carnot's theorem states that, given any triangle ABC, the signed sum of perpendicular distances from the circumcenter O to the sides (i.e., signed lengths of the pedal lines from O) is OO_A+OO_B+OO_C=R+r, where r is the inradius and R is the circumradius.
Carnot's theorem (inradius, circumradius) (Wikipedia)
In Branch: Geometry, Lead sentenceQuote, In Branch: Geometry, Lead sentence
In Euclidean geometry, Carnot's theorem (English: kar-NOH, French: [kaʁno]) states that the sum of the signed distances from the c
Carnot's Theorem (Wikipedia)
Proved By: Lazare Carnot, LeadQuote, Proved By: Lazare Carnot, Lead
The theorem is named after Lazare Carnot (1753-1823)
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