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The isotomic disc and the isogonal disc conjecture

Published online by Cambridge University Press:  15 October 2025

Martin Lukarevski ,
Stanley Rabinowitz  and
J. A. Scott
Martin Lukarevski
Affiliation:
Department of Mathematics and Statistics, University Goce Delcev Stip, North Macedonia e-mail: martin.lukarevski@ugd.edu.mk
Stanley Rabinowitz
Affiliation:
1 Shiptons Lane, Great Somerford, Chippenham SN15 5 EJ
J. A. Scott
Affiliation:
545 Elm St Unit 1, Milford, NH 03055, USA e-mail: stan.rabinowitz@comcast.net
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Extract

Let ABC be a triangle with incentre I, circumcentre O, orthocentre H, centroid G and symmedian point K. In standard notation, the triangle ABC has sides a, b, c, semiperimeter s, circumradius R and inradius r. Euler’s well-known result that the incentre I is always within the orthocentroidal disc DGH, the disc with diameter GH, is probably the first result about the location of the incentre in some disc formed by triangle centres. Investigating the location of the incentre I in other discs, in [1] we proved that the incentre is interior to the Brocard disc DOK, that is, the disc with diameter OK. The disc is named after the French military meteorologist and geometer Henri Brocard (1845-1922), known in triangle geometry for the Brocard points and the Brocard angle (see [2]).

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Type
Articles
Information
The Mathematical Gazette , Volume 109 , Issue 576 , November 2025 , pp. 496 - 500
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Copyright
© The Authors, 2025 Published by Cambridge University Press on behalf of The Mathematical Association

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References

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