# Talk:Inscribed figure

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Field: Geometry

## Definition

Until recently, the article said

Specifically, at all points where figures meet, their edges must lie tangent. There must be no object similar to the inscribed object but larger and also enclosed by the outer figure.

The former sentence was removed as redundant. Actually, the former sentence is correct, and the latter sentence is not conventional: For example, a triangle has three inscribed squares (obviously all similar to each other), but in general they are not the same size; the smaller ones still count as inscribed.

Moreover, the article says

Of these figures, an inscribed one is a figure of maximal size among those of the same shape enclosed by X. Usually it is unique in size....

This is self-contradictory, since of course the maximal size is unique.

So I'm going to alter it take these observations into account. Duoduoduo (talk) 21:39, 9 March 2011 (UTC)

Some issues with the re-instated sentence:
1. Circles have no edges, so for figures inscribed in a circle, and for circles unscribed in some figure, the statement about edges lying tangent is meaningless.
2. For the maximal isosceles triangle whose base and height have equal length that fits in a given square, the two long sides (of length ½√5 times the height) of the triangle meet one side of the square at an angle in the middle; nothing tangent there, so the statement is false.
But you are right that conventionally maximality is not required. For an inscribed polygon, according to MathWorld, a triangle inscribed in a second triangle has each of its vertices lying on a side of the second triangle,[1] and an inscribed circle of a polygon is a circle that is tangent to each of the polygon's sides,[2] while bymath.com states that all vertices of a polygon inscribed in a circle lie on the circle.[3] (Unfortunately, these are not reliable sources in the sense of Wikipedia.) It is not clear that there is a truly unifying definition for arbitrary combinations of geometric figures, and it may be best to restrict ourselves to the traditional cases in which one of the two figures is a polygon and the other a circle or polygon, until an authoritative source is found offering further or more general definitions.  --Lambiam 23:15, 9 March 2011 (UTC)
Mathworld is considered a reliable source and is cited in many Wikipedia articles. The bymath site looks reliable to me.
I've reworded the sentence completely to meet your objections (omitting "edge" which was used in a non-technical sense of "perimeter", and deleting "tangent" which I think was properly used to mean "touching but not crossing"). The current version may still be insufficient, however, since I think it does not preclude a polgon "inscribed" in a circle with one of the vertices not touching. But at least the current version is closer than anything yet. Duoduoduo (talk) 23:52, 9 March 2011 (UTC)
Thanks, but I'm afraid the definition is actually still quite a mess (for which I accept a large part of the responsibility).
Some investigations show the following:
1. "Figure F is inscribed in figure G" means precisely the same as "figure G is circumscribed about figure F"
2. The definite article in "the inscribed/circumscribed circle" (of a polygon), or "the inscribed square" (of a circle), is justified because for these specific cases the definition implies uniqueness. In general it does not; there are many non-similar triangles inscribed in a given circle.
3. Maximality or immobility are red herrings. Similar triangles of different sizes can be inscribed in an ellipse. An obtuse triangle inscribed in a circle can be translated while keeping it inside.
4. We have articles Circumscribed circle, Circumscribed sphere, Inscribed angle, Inscribed circle (redirects to Incircle and excircles of a triangle, which is insufficiently general), Inscribed sphere, and Inscribed square problem (which contains an implicit definition of "inscribed square" of an arbitrary plane simple curve).
I don't know what the best immediate course of action is, but in any case I've slapped an {{accuracy}} tag on the article for now.  --Lambiam 11:11, 10 March 2011 (UTC)
The article Inscribed square problem makes it seem difficult for there to be a single generic definition, since it shows a graph in which the inscribed square lies partly outside the other figure.
You're right that the present version is still no good. How about if I revise it as a stop-gap measure to include the following sentences: (1) your sentence "Figure F is inscribed in figure G" means precisely the same as "figure G is circumscribed about figure F"; (2) "A circle or ellipse inscribed in a convex polygon is tangent to every side of the polygon and completely inside it"; and (3) "a polygon inscribed in a circle, ellipse, or convex polygon is completely inside it and has each vertex on the outer figure". Duoduoduo (talk) 15:43, 10 March 2011 (UTC)
For both (2) and (3) the clause "is completely inside it" and the restriction to convex polygons are consequences of the rest of these definitions. If you simplify the definitions by only requiring in (2) that the circle/ellipse is tangent to every side, and in (3) that each vertex of the polygon lies on the outer figure, and leaving out the word "convex", it is a corollary that convex curves, touching all edges without crossing, can only be inscribed in polygons that are also convex, and only convex polygons can be inscribed in convex curves. The proofs are almost trivial. It follows then that in each case F is contained in G. (In the general case, for not necessarily convex figures, it still follows that the convex hull of F is contained in the convex hull of G.) The MathWorld and bymath.com definitions linked to above, and others I found using Google book search, don't have an inside clause or convexity restriction as part of the definitions. That then also makes the inscribed-square problem fit in. I also feel uneasy about the polygon-in-polygon case, which is not covered by the sources I saw except for triangle-in-triangle, but then it was also required that each side of the outer triangle is touched by a vertex of the inner one, so perhaps leave that out for now. For the rest this looks good as a stopgap.  --Lambiam 23:56, 10 March 2011 (UTC)