Harcourt's theorem is a formula in geometry for the area of a triangle, as a function of its side lengths and the perpendicular distances of its vertices from an arbitrary line tangent to its incircle.[1]
The theorem is named after J. Harcourt, an Irish professor.[2]
Let a triangle be given with verticesA, B, and C, opposite sides of lengths a, b, and c, area K, and a line that is tangent to the triangle's incircle at any point on that circle. Denote the signed perpendicular distances of the vertices from the line as a ', b ', and c ', with a distance being negative if and only if the vertex is on the opposite side of the line from the incenter. Then
Degenerate caseedit
If the tangent line contains one of the sides of the triangle, then two of the distances are zero and the formula collapses to the familiar formula that twice the area of a triangle is a base (the coinciding triangle side) times the altitude from that base.
Extensionedit
If the line is instead tangent to the excircle opposite, say, vertex A of the triangle, then[1]: Thm.3
Dual propertyedit
If rather than a', b', c' referring to distances from a vertex to an arbitrary incircle tangent line, they refer instead to distances from a sideline to an arbitrary point, then the equation
^ abDergiades, Nikolaos; Salazar, Juan Carlos (2003), "Harcourt's theorem" (PDF), Forum Geometricorum, 3: 117–124, MR 2004117.
^G.-M., F. (1912), "Théorème de Harcourt", Exercises de géométrie: comprenant l'exposé des méthodes géométriques et 2000 questions résolues, Cours de mathématiques elementaires (in French) (5th ed.), Maison A. Mame et fils (Tours) & J. de Gigord (Paris), p. 750.
harcourt, theorem, formula, geometry, area, triangle, function, side, lengths, perpendicular, distances, vertices, from, arbitrary, line, tangent, incircle, theorem, named, after, harcourt, irish, professor, contents, statement, degenerate, case, extension, du. Harcourt s theorem is a formula in geometry for the area of a triangle as a function of its side lengths and the perpendicular distances of its vertices from an arbitrary line tangent to its incircle 1 The theorem is named after J Harcourt an Irish professor 2 Contents 1 Statement 2 Degenerate case 3 Extension 4 Dual property 5 ReferencesStatement editLet a triangle be given with vertices A B and C opposite sides of lengths a b and c area K and a line that is tangent to the triangle s incircle at any point on that circle Denote the signed perpendicular distances of the vertices from the line as a b and c with a distance being negative if and only if the vertex is on the opposite side of the line from the incenter Then aa bb cc 2K displaystyle aa prime bb prime cc prime 2K nbsp Degenerate case editIf the tangent line contains one of the sides of the triangle then two of the distances are zero and the formula collapses to the familiar formula that twice the area of a triangle is a base the coinciding triangle side times the altitude from that base Extension editIf the line is instead tangent to the excircle opposite say vertex A of the triangle then 1 Thm 3 aa bb cc 2K displaystyle aa prime bb prime cc prime 2K nbsp Dual property editIf rather than a b c referring to distances from a vertex to an arbitrary incircle tangent line they refer instead to distances from a sideline to an arbitrary point then the equation aa bb cc 2K displaystyle aa prime bb prime cc prime 2K nbsp remains true 3 p 11 References edit a b Dergiades Nikolaos Salazar Juan Carlos 2003 Harcourt s theorem PDF Forum Geometricorum 3 117 124 MR 2004117 G M F 1912 Theoreme de Harcourt Exercises de geometrie comprenant l expose des methodes geometriques et 2000 questions resolues Cours de mathematiques elementaires in French 5th ed Maison A Mame et fils Tours amp J de Gigord Paris p 750 Whitworth William Allen Trilinear Coordinates and Other Methods of Modern Analytical Geometry of Two Dimensions Forgotten Books 2012 orig Deighton Bell and Co 1866 http www forgottenbooks com search q Trilinear coordinates amp t books Retrieved from https en wikipedia org w index php title Harcourt 27s theorem amp oldid 986761188, wikipedia, wiki, book, books, library,
