Talk:Homothetic center

Figure 3 homothetic center of two circles

The diagram for figure 3 includes the variable d, which I assume is distance. However the diagram does not indicate d, so it is incomplete in this regard.24.76.167.34 (talk) 16:15, 3 December 2009 (UTC)

Mistake

There is a mistake in the paragraph "Pairs of antihomologous points lie on a circle". The triangles mentioned there ARE NOT SIMILAR! (the picture on the left is also WRONG!) Steff --78.157.4.236 (talk) 23:19, 26 October 2010 (UTC)

I agree with Steff. If you were to select P such that α = π/2, the line EP would intersect the green circle as a chord, but α = π/2 on the blue circle would have that equivalent line EQ intersecting that circle at a tangent. Clearly not consistent with the construction. -- Tom N (tcncv) talk/contrib 01:05, 27 October 2010 (UTC)

Event with the vertices rearranged, the triangles in the Pairs of antihomologous points lie on a circle section are not mathematically similar. They are visually close, but that is misleading). I have added back a more prominent factual accuracy dispute notice. I'm also asking for an opinion from the reference desk.

As another counterexample, consider if the two circles are superimposed with a common center C=D and the same radius. In this case, ∠QPC and ∠QCE are both equal to β. However, the angles ∠CQP and ∠QPC are identical by symmetry, so ∠CQP also equals β. Now we have ∠CQP equal to ∠QCE, which implies that the lines QP and CE are parallel. This conflicts with general construction case were these lines intersect at E. -- Tom N (tcncv) talk/contrib 04:24, 28 October 2010 (UTC)

α and β as drawn at points Q and P satisfy α_Q + β = π while α and β as drawn at points C and P satisfy α_C + β + θ = π. So α_Q ≠ α_C . Bo Jacoby (talk) 07:23, 28 October 2010 (UTC).

Thanks for the input. Not sure if what if anything is salvageable from this section. It seems clear that the first part is wrong. Not sure about the radical axis material has any relevance. Looks like most of the material came from editor. I'll ask them to take a look. -- Tom N (tcncv) talk/contrib 02:12, 29 October 2010 (UTC)

I think that the proof is wrong; but the end result seems quite correct.

The chord theorem does say that $d_{EP}d_{EQ}=d_{ES}d_{ET}$. This is usually proven by means of similar triangles; but not by those suggested in the article, which indeed are not similar. The article could be fixed by inserting the correct proof for this instance of the chord theorem; but I wonder if this really is necessary. Those who could follow the proof, also ought to be able to follow a reference.

The only trouble seems to be that for some reason the classical chord theorem (found in Euclid) is given rather implicit in our article, and as if it were a corollary of a somewhat different formulation by Jacob Steiner. This might motivate a revision of the "power of a point"/"chord theorem" article(s); but I do not think that the "Homothetic center" article is the best place to provide the proof.

If you still wish to include the proof of the specific instance of the chord theorem used, you should introduce homologous intersections between the line through EC and the two circles; let us call them U and V (so that EUCDV come in this order in the figure). Then, EPU is similar to EVQ; the reason being that P and and V are periphery angles on similar circle sectors. Continue from there, mutatis mutandis (and modulo the printing errors I may have provided...).

I think that the original author mixed up the diameter and the tangent square cases of the chord therorem. JoergenB (talk) 21:35, 29 October 2010 (UTC)

I've added the preceeding section to the troubled ones. The claim about the angles α and β, which as Bo Jacoby pointed out contains errors, was made already there. This mistake then propagates to the next section.

Still, I think it is a nice enough article, and with good illustrations. User Tom N asked User:WillowW for input; I agree, and think that WillowW actually would be the most apt person to correct this, especially the images. JoergenB (talk) 16:32, 30 October 2010 (UTC)

My deepest apologies for having made such a mistake and I'll correct it as soon as I can. I was working from a textbook, but the proof given was so cursory that I had to fill in the details. With humility, Willow (talk) 16:03, 31 October 2010 (UTC)

Only by doing nothing can we avoid doing mistakes... This really is a nice article, and I do think the errors should be reasonably easy to fix.

I'm not very used to the svg format; supposedly, one should use another editor than source-file-emacs in order to fix it; but if it is of any use, check the modification of your Figure 4 .

The two angles marked α stand on similar circle sectors, and therefore are of equal size (by a slight extension of the periphery angle theorem). The angle θ is shared and the third angles thus trivially are of equal size. (PS: I'm not used to image uploading, and may have handled the copyrights wrong. I found no alternative : "A derived work of one already present and licensed at commons". If the license is wrong, or if you have no use for the file, I can correct or remove either.) JoergenB (talk) 16:39, 31 October 2010 (UTC)

Internal v External centers: language

The article states internal centers have opposite sense (chirality) while external centers have the same sense - which is a definition (of sorts). However 'internal' and external' language doesn't help much in describing the points themselves because (for instance) the orthocenter of a triangle is an external homothetic center of the nine point and circumcircles but lies interior to both. Could it not be stressed that the 'internal'/'external' distinction does not relate the placment of the points/centers to the objects themselves? — Preceding unsigned comment added by 78.146.2.128 (talk) 13:24, 23 February 2016 (UTC)