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Some geometry problems ( like this and this ) have short solutions if we use involutions. What references are there for solving geometry problems using involutions? I am particularly interested in theoretical aspects of involutions, and maybe a reference with solved problems like these two presented in the links.

There are quite a few problems on AoPS that are solved using involutions. I searched there for help, but it seems that the user that posted most of the solutions (grobber) is not active anymore. There was a suggestion to make a teaching article for such problems but there was no one to finish the task. This question was on StackExchange for almost a month now without any answers or comments, that's why I've posted it here.

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    $\begingroup$ Not a real/specific question. $\endgroup$
    – Martin Brandenburg
    Commented Jul 13, 2011 at 14:52
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    $\begingroup$ is involution the same as inversion in a circle? many titles store.doverpublications.com/0486434761.html $\endgroup$
    – Will Jagy
    Commented Jul 13, 2011 at 18:48
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    $\begingroup$ Is the tag a joke? $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Jul 13, 2011 at 20:40
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    $\begingroup$ Mariano, Darij is (or maybe was) a presence on AoPS, asked the first linked question in Beni's question, and would surely know the meaning of the jargon. So I put in the tag, having no better way to ask him to participate. This is the second such tag I have made, the first could be said to be by request (but not specifically to me), see mathoverflow.net/questions/69542/uniformly-convex-spaces I understand that these tags will be deleted in some monthly maintenance Anton does. $\endgroup$
    – Will Jagy
    Commented Jul 13, 2011 at 20:56
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    $\begingroup$ It is also a joke. $\endgroup$
    – Will Jagy
    Commented Jul 13, 2011 at 20:56

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I printed out your two link problems and looked at the language. It becomes clear that the word "involution" is used by "grobber" as any instance of a projective transformation that returns to the identity when repeated. This would include reflection in a fixed line, inversion across a fixed circle, rotation of 180 degrees around a fixed point (this one is called reflection in the point, evidently). The other language on pencils of lines and the like makes it clear that we are talking about projective geometry.

The attitude that is somewhat new to me is the statement that "there is one and only one of these mappings that accomplishes ----, further work shows that the particular mapping is __." The more familiar application is in Moebius transformations, where any three distinct given points in $\mathbb C \cup \infty$ can be sent to any other three points, then the particular transformation that does it can be worked out if necessary.

So, I am convinced "involution" is a collective word, as in "Moebius transformations." The Dover titles I mentioned include some that emphasize elementary problems. I can also point out that Coxeter was very fond of projective techniques, and some of his books are available.

EDIT: I found only a few other mentions of the word "involution" on AoPS that clearly relate to projective geometry. The most useful was about the Desargues Involution Theorem. This is shown, in detail and in somewhat outdated language, in section 63,pages 265-273, of a Dover reprint (and translation), Heinrich Dorrie, 100 Great Problems of Elementary Mathematics.

http://store.doverpublications.com/0486613488.html

I also found plenty of lesson material online, professors writing their own versions of the theorem. Here is the one that sent me to Dorrie

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  • $\begingroup$ A projective transformation of the complex line is any function from the complex numbers plus infinity to the complex numbers plus infinity that preserves the complex cross ratio (i.e. plug in complex numbers in the cross ratio formula). This does NOT include reflections about a fixed line or inversions because they conjugate the cross ratio. Up to a reasonable choice of coordinate system, all involutions of this kind are $z$, $-z$ and $1/z$. Since the first two already have other names, this is really about $1/z$ after an appropriate choice of origin, axis and scaling. $\endgroup$
    – user11235
    Commented Aug 23, 2020 at 9:06
  • $\begingroup$ And yes, the projective transformations of the complex line are exactly the Möbius transformations. $\endgroup$
    – user11235
    Commented Aug 23, 2020 at 9:10
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You say you are interested in theoretical aspects as well as seeing solved problems.I believe picking up any Russian book on geometry would do the trick, but I personally prefer the series of books "Geometric Transformations I-IV" by I.M. Yaglom (which have been translated to English). The topics in the four books are divided as isometries, similarities, affine and projective transformations, and conformal transformations. There are quite a few problems in these books and they can be quite challenging, in particular I believe it does a great job at helping one think in terms of synthetic solutions versus analytical approaches. If after that you would like to try your hands at more problems, you can look at the relevant chapters in Prasolov's "Problems in plane and solid geometry".

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My advise is Theory and Problems of Projective Geometry in the Schaum's Outline Series, written by Frank Ayres. It is indeed a very nice method of solving problems in geometry.

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