okay this one is from Prof. Mircea Lascu from Zalau, Romaniaand Prof. V. Cartoaje from Ploiesti, Romania. It goes like this: given being a triangle ABC for every point M inside we construct the points AM, BM, CM on the circumcircle of the triangle ABC such that A, AM, M are collinear and so on. Find the locus of these points M for which the area of the triangle AM BM CM is bigger than the area of the triangle ABC.
Problem
Source:
Tags: geometry, circumcircle, ratio, Really old, geometry unsolved, algebra
Valentin Vornicu
14.03.2003 00:29
so I guess no one is considering this problem to be a challenge ?
iandrei
14.03.2003 14:34
Well , as far as I know this is an open problem, but I haven't tried it yet either. Doesn't look like a piece of cake although.
kappa
17.04.2003 22:35
ok i found an article witch, after tons of calculations, comes up with S'/S= (get ready) prod ( a^2+b^2xy+c^2y ) / (x(x+1)b^2+(x+1)c^2-a^2x) where a,b,c the sides, and x,y,z the ratios BD/CD , with xyz=1; gotta go bye PS i dont think this will help a lot, cause i guess valentin knows this article. its gazeta matematica no 3 from 2002
Valentin Vornicu
17.04.2003 23:42
yes ... i know that ... and indeed it doesn't help at all ... there is another article concerning these types of triangles in the Math Gazette 2/2003 ( just bought it today! ) - i will study it and if I find anything intersting, or worth mentioning I will post it
Myth
17.04.2005 16:56
I think the question will be quite meaningless after you will see that picture I constructed using SketchPad. It is a high-precise shape of the required locus. So it is not an any traditional line. And I have some doubts it have a nice parametrization.
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mecrazywong
17.04.2005 17:01
The locus just looks like a monster...I'm glad I didn't try this problem before
darij grinberg
17.04.2005 17:40
Myth wrote: I think the question will be quite meaningless after you will see that picture I constructed using SketchPad. It is a high-precise shape of the required locus. So it is not an any traditional line. The curve bordering the locus (i. e. the set of all points M such that the area of triangle $A_MB_MC_M$ equals to the area of triangle ABC) is a sextic. See Hyacinthos messages #6178 and #6179. So, indeed, the question is meaningless. However, http://www.mathlinks.ro/Forum/viewtopic.php?t=21900 gives an assertion about a part of this locus. Darij
Valentin Vornicu
17.04.2005 19:04
Myth wrote: I think the question will be quite meaningless after you will see that picture I constructed using SketchPad. It is a high-precise shape of the required locus. So it is not an any traditional line. And I have some doubts it have a nice parametrization. Thanks Mikhail for clearing this out after exactly two years
Valiowk
17.04.2005 19:08
Valentin Vornicu wrote: Thanks Mikhail for clearing this out after exactly two years I think you mean two years and two months, don't you, Valentin?
Valentin Vornicu
17.04.2005 19:25
Valiowk wrote: I think you mean two years and two months, don't you, Valentin? I ment two years since the last message on the thread : http://www.mathlinks.ro/Forum/viewtopic.php?p=126#p126
orl
26.04.2005 03:41
Valentin Vornicu wrote: Valiowk wrote: I think you mean two years and two months, don't you, Valentin? I ment two years since the last message on the thread : http://www.mathlinks.ro/Forum/viewtopic.php?p=126#p126 Looking at the time I would say less than two years.