459:.) 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. --
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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
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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
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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
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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
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in the sense of
Knowledge.) 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
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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".
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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
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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
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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.
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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.
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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.
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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....
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The definite article in "the inscribed/circumscribed circle" (of a polygon), or "the inscribed square" (of a circle), is justified because
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For the maximal isosceles triangle whose base and height have equal length that fits in a given square, the two long sides (of length
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Thanks, but I'm afraid the definition is actually still quite a mess (for which I accept a large part of the responsibility).
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the definition implies uniqueness. In general it does not; there are many non-similar triangles inscribed in a given circle.
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Mathworld is considered a reliable source and is cited in many
Knowledge articles. The bymath site looks reliable to me.
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Maximality or immobility are red herrings. Similar triangles of different sizes can be inscribed in an ellipse. An
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Specifically, at all points where figures meet, their edges must lie tangent. There must be no object
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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and an inscribed circle of a polygon is a circle that is tangent to each of the polygon's sides,
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while bymath.com states that all vertices of a polygon inscribed in a circle lie on the circle.
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I don't know what the best immediate course of action is, but in any case I've slapped an
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authoritative source is found offering further or more general definitions. --
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to the inscribed object but larger and also enclosed by the outer figure.
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This is self-contradictory, since of course the maximal size is unique.
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inscribed in a circle can be translated while keeping it inside.
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So I'm going to alter it take these observations into account.
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