Talk:Power of a point

The chord theorem

The chord theorem is classical. It may be found in Euclid, but not in enwp; Chord theorem redirects to this article. The "power of a point" approach depends on plane geometry being identified with "geometry on R²", while the common proofs of the classical theorem depends on an efficient use of similar triangles.

Given the one, and the appropriate definitions, you may deduce the other as a corollary. In the present article, the chord theorem is mentioned as a corollary to the power of a point existence. I do not think this is a good way to do it. Either we should have two articles, cross referring to each others; or we should have one article, named Chord theorem, and include the power of a point discussion as a corollary; or even as a part of a section named Modern alternative approaches, or something to that effect.

As it is now, it is hard to refer to the (rather fundamental!) chord theorem from other articles, since it refers to this article, which really isn't very clear on that subject. JoergenB (talk) 13:35, 30 October 2010 (UTC)[reply]

The definition here seems pedagogically/conceptually/historically backward

The way Steiner came up with the 'power' of a point was by looking at the tangent-secant theorem (and its trivial corollary the intersecting secants theorem) and intersecting chords theorem from Euclid, and noticing that they implied the existence of some constant product of lengths for any chosen point and circle. It seems like this should be the fundamental definition. That this constant is equal to the difference of the squares of the distance to the center and the radius is a consequence.

Making the difference of squares the basic definition may be more convenient for calculations, but misses the point as far as explaining how this concept arose or why it is useful. –jacobolus (t) 19:26, 17 January 2023 (UTC)[reply]

One could insert a section considering Steiner's original introduction of the power of a point (Crelle-Journal 1, p. 163), which is coordinate (analytic geometry) free, relying on the tangent secant theorem and intersecting secants theorem. I think, the introduction given in the article using anal. geom. is in literature the most common one. Ag2gaeh (talk) 10:49, 19 January 2023 (UTC)[reply]

Thanks for the reference. Here is a good-quality scan:

Steiner, Jakob (1826). "Einige geometrischen Betrachtungen" [Some geometric considerations]. Crelle's Journal (in German). 1: 161–184. doi:10.1515/crll.1826.1.161. Figures 8–26.

The reason I think it is pedagogically useful to lead with intersecting secants/chords is that that gives (in and of itself) a strong motivation for considering this constant number as summarizing the relationship of the point to the circle: "look here, student, whatever cutting line we choose this number is always the same"; the purely numerical/algebraic version initially seems like an arbitrary formula. –jacobolus (t) 17:22, 19 January 2023 (UTC)[reply]