--Sig_/P62dl/pCrNyOhSYjGYTSiaP Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: quoted-printable * Horea Raducan, 2008-02-16, 03:06: > If one of the points is on the enclosing circle, is that a valid > solution ? This is necessarily so if the task is to make sense. Assume you have two points A and B and construct M such that M is the middle of AB. Then the smallest circle that encloses A, M, and B should be the one that has M as its center and has a diameter equal to the length of AB because otherwise you run into the fundamental issue that the infimum of the set of the diameters of all circles that enclose A, B, and M is the length of AB while a circle with precisely this diameter does not meet the requirements. In other words: Yes, it would make no sense not to allow this as a valid solution because it is *so* easy to construct a case in that this definition runs into fundamental trouble. Just in case: The fundamental issue is that assuming the solution 'point on circle' is invalid would mean that the very basic case of three equidistant points on a straight line had no valid solution at all. While mathematics is no natural science you will find very few cases in which the definitions (like that of 'enclosing circle' in this case) AIM AT being incompatible with common sense. Just to give an example of alien-looking definitions that make perfect sense: The field being defined by the set {0,1} with multiplication and addition defined as 0*0 = 0*1 = 1*0 = 0, 1*1 = 1 0+0 = 1+1 = 0, 0+1 = 1+0 = 0 This can be interpreted as binary digits with the operations AND (as multiplication) and XOR (as addition). Take a look at this table: a b | c s ----|----- 0 0 | 0 0 0 1 | 0 1 1 0 | 0 1 1 1 | 1 0 a and b are the inputs, c is the result of AND and s is the result of XOR. Read 'c' as 'carry' and 's' as 'sum' and you see an addition of two bits with carry. Hope that was not to much math hacking :) Josef 'Jupp' Schugt -- Blog available at http://cip.physik.uni-bonn.de/~jupp/ PGP key with id 6CC6574F available at http://wwwkeys.de.pgp.net/ Jabber - http://www.jabber.org/ - contact information on request --Sig_/P62dl/pCrNyOhSYjGYTSiaP Content-Type: application/pgp-signature; name=signature.asc Content-Disposition: attachment; filename=signature.asc -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.4.7 (GNU/Linux) iD8DBQFHtd/Grhv7B2zGV08RAoYEAJ9E/UofZ6KIoRPnVlwBFKinHxy1WwCdHuhM BEU3XGNJLYHFv7/L6uD5m2s zR -----END PGP SIGNATURE----- --Sig_/P62dl/pCrNyOhSYjGYTSiaP--