Isogonal and Isotomic Conjugalities: What is this about?
A Mathematical Droodle


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Try to arrange the points on the side lines of a triangle so that they are either collinear or their cevians concur.

Explanation

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Copyright © 1996-2017 Alexander Bogomolny

It's a well known fact that either isogonal or isotomic conjugates of concurrent cevians are also concurrent. In both cases the result easily follows from Ceva's theorem. The points of concurrency are said to be isogonal (or isotomic, as the case may be) conjugates of each other.

Strangely, a similar fact with a virtually identical derivation but on the basis of Menelaus' theorem is much less known. I do not believe that the ensuing correspondence has even been acknowledged with a distinctive name.

So here it is: if three points - one on each side line of a triangle - are collinear, so are their isogonal and isotomic conjugates. (The isotomic case has been discussed by R. Honsberger.)

References

  1. R. Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, MAA, 1995, pp. 153-154

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Copyright © 1996-2017 Alexander Bogomolny

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