One Theorem Kills It All
Let f ( x ) = x 3 + 6 x 2 + 3 x + 7 and let g ( x ) = x 2 + 4 x + 1 . The roots of f ( x ) and g ( x ) are plotted on the complex plane. A triangle is drawn with vertices at the roots of f ( x ) and an ellipse is drawn with foci at the roots of g ( x ) that is tangent to the triangle at the midpoint of the side that connects the two complex roots. Find the number of intersection points of the triangle and ellipse.
Details and Assumptions
- The tangency point counts as an intersection point.
The answer is 3.
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1 solution
Obscure. Don't like.
Even if you had said that g ( x ) was the derivative of f ( x ) , there is a high probability that nobody will use the theorem.
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This problem was just my way of sharing the theorem.
Exactly!!!!!!
Yeah I don't like having to try to reprove age-old theorems, but you must admit it is always nice to learn new stuff. Who knows, not learning Marden's might come back to bite you some day. :D
Oh, thanks for the interesting reference to Marden's Theorem. The answer had to be 5, 3, or 1, and I had 3 tries.
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Yep that's what most people did. I considered making it multiple choice but to me it doesn't really matter. I just wanted to share the theorem.
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It already inspired another problem, and one solver used Marden's Theorem
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@Michael Mendrin – When I saw that problem I knew it could be solved through mardens theorem, although I never got around to bashing through the calculations. I'm glad someone else has used it though :D
@Michael Mendrin – Sir, if it's not too much trouble, can you outline a proof of the Steiner inellipse having maximum area? The ones that I have seen use affine transforms. Is there another way of proving it?
I don't like how a geometric theorem was used in algebra.
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I like it when a geometric theorem gets used in other math branches like algebra. That's the whole spirit of doing mathematics, "borrowing" concepts and theorems from other branches to solve or prove things.
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@Michael Mendrin – I guess but shouldn't it be under geometry?
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@Sal Gard – Not necessarily. For example, in Topology, we have the "Fixed Point Theorem" that has been used to solve problems elsewhere, like differential equations. We wouldn't call the branch of differential equations "Topology".
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@Michael Mendrin – Good to know. I appreciate your consideration to my ignorance.
@Michael Mendrin – I am not trying to get answers but I am curious to learn more about the ideas employed in geometry problems such as Xuming Liang's recent circles problem. I would if you could suggest any good ones.
As for the one posted here, well, I see your point about whether it should have been classified as a geometry problem here in Brilliant, instead of algebra. I don't know if there's a good answer for that. There are other such problems here that strongly involve both geometry and algebra.
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@Sal Gard – If you search "Xuming Liang" for Problems, you'll find many more posted by him. I would say that most of his problems are quite difficult problems in classical geometry.
Nice problem about Marden's Theorem. I will add this to my list of theorems! Thank you very much!
By Marden's Theorem , the ellipse is tangent to the triangle at the midpoints of each side, so there are 3 intersection points.