In geometry, the isotomic conjugate of a point P with respect to a triangle △ABC is another point, defined in a specific way from P and △ABC: If the base points of the lines PA, PB, PC on the sides opposite A, B, C are reflected about the midpoints of their respective sides, the resulting lines intersect at the isotomic conjugate of P.
We assume that P is not collinear with any two vertices of △ABC. Let A', B', C' be the points in which the lines AP, BP, CP meet sidelines BC, CA, AB (extended if necessary). Reflecting A', B', C' in the midpoints of sides BC, CA, AB will give points A", B", C" respectively. The isotomic linesAA", BB", CC" joining these new points to the vertices meet at a point (which can be proved using Ceva's theorem), the isotomic conjugate of P.
Coordinatesedit
If the trilinears for P are p : q : r, then the trilinears for the isotomic conjugate of P are
where a, b, c are the side lengths opposite vertices A, B, C respectively.
Propertiesedit
The isotomic conjugate of the centroid of triangle △ABC is the centroid itself.
Isotomic conjugates of lines are circumconics, and conversely, isotomic conjugates of circumconics are lines. (This property holds for isogonal conjugates as well.)
isotomic, conjugate, geometry, isotomic, conjugate, point, with, respect, triangle, another, point, defined, specific, from, base, points, lines, sides, opposite, reflected, about, midpoints, their, respective, sides, resulting, lines, intersect, isotomic, con. In geometry the isotomic conjugate of a point P with respect to a triangle ABC is another point defined in a specific way from P and ABC If the base points of the lines PA PB PC on the sides opposite A B C are reflected about the midpoints of their respective sides the resulting lines intersect at the isotomic conjugate of P Contents 1 Construction 2 Coordinates 3 Properties 4 See also 5 References 6 External linksConstruction edit nbsp We assume that P is not collinear with any two vertices of ABC Let A B C be the points in which the lines AP BP CP meet sidelines BC CA AB extended if necessary Reflecting A B C in the midpoints of sides BC CA AB will give points A B C respectively The isotomic lines AA BB CC joining these new points to the vertices meet at a point which can be proved using Ceva s theorem the isotomic conjugate of P Coordinates editIf the trilinears for P are p q r then the trilinears for the isotomic conjugate of P are a 2 p 1 b 2 q 1 c 2 r 1 displaystyle a 2 p 1 b 2 q 1 c 2 r 1 nbsp where a b c are the side lengths opposite vertices A B C respectively Properties editThe isotomic conjugate of the centroid of triangle ABC is the centroid itself The isotomic conjugate of the symmedian point is the third Brocard point and the isotomic conjugate of the Gergonne point is the Nagel point Isotomic conjugates of lines are circumconics and conversely isotomic conjugates of circumconics are lines This property holds for isogonal conjugates as well See also editIsogonal conjugate Triangle centerReferences editRobert Lachlan An Elementary Treatise on Modern Pure Geometry Macmillan and Co 1893 page 57 Roger A Johnson Advanced Euclidean Geometry Dover 2007 ISBN 978 0 486 46237 0 pp 157 159 278External links editWeisstein Eric W Isotomic Conjugate MathWorld Pauk Yiu Isotomic and isogonal conjugates Navneel Singhal Isotomic and isogonal conjugates Retrieved from https en wikipedia org w index php title Isotomic conjugate amp oldid 1127496233, wikipedia, wiki, book, books, library,
