I have solved the
with Prime 6 or Mathcad 15 - see pls attach
Can you solve it more sort, more nice 😉
From Wikipedia, the free encyclopedia In geometry, the Malfatti circles are three circles inside a given triangle such that each circle is tangent to the other two and to two sides of the triangle. They are named after Gian Francesco Malfatti, who made early studies of the problem of constructing
There is a long-serving regular contributor here in the forum who likes to comment on such occasions that a solution without animation would not be a solution.
You may have read it here once or twice (or was it 100 times?). 😈
I apologize as this is neither a particularly beautiful nor a particularly useful animation.
Yes, not each animation is informative and useful.
I hope this is more interesting - an animation of the numeric solution
one connected problem.
Let's wait for other decisions, and then open our playing cards, sorry, our solutions.
Here you can compare the percentage of the triangle area covered by the Malfatti circles and the true solution of the Malfatti problem (maximum covered area by three circles). It demonstrates that the Malfatti circles never are solutions to the Malfatti problem.
@ValeryOchkov wrote:
Super! Can you show Mathcad-sheets?
Sure, there is nothing confidential or secret in my sheet.
Here you are. The MC15 file is attached. Have fun!
And what about a tetrahedron and three spheres?
Why just three? Wouldn't four be more appropriate? Looking forward to see your solution 😉
I guess that the solution for this 3D Malfatti problem (maximize the volume of the spheres) would be easier to solve (greedy algorithm) than the 3D generalization of the Malfatti circles (the spheres all being tangent to each other and to the tetrahedron sides). Proving that a greedy algorithm gives the solution for the maximum volume would be another thing. The prove for the 2D problem is 25 pages and also involves computer calculations to deal with all relevant constellations.
In case you are really interested in a high dimensional generalization of Malfatti's problem, you may check this
https://www.aimsciences.org/article/exportPdf?id=c026650a-cc3b-4ec0-9a6a-d729ccf92bdc
or this
http://www.aimsciences.org/article/id/0953c355-7831-43ea-9412-0fdcb8f9b5a7
A classic generalization of the Malfatti circles is to change the straight triangle sides for circle arcs.
I now found an interesting paper where the Malfatti circles were generalized (still in 2D) in a new way. The task is to fit six circles into the triangle, tangent to each other and to the triangle sides. Of course "tangent to each other" can't mean that every circle is in touch with each of the remaining ones as is the case with the original 3 Malfatti circles. While the original three Malfatti circles could be found solely by analytic methods (in my sheet I didn't as Mathcads symbolic would not be able to solve the necessary equations), the solution for the six circles needs to make heavily use of numerical methods. In case you have too much spare time (😉), here is the link: https://www.sciencedirect.com/science/article/pii/S0925772112000259
Good luck!!
I solved with students the problem of how a triangle fits into a circle using numerical methods - see animation above. The idea came to inscribe two, three, etc. circles in a triangle. The problem was quickly solved numerically (see pictures bellow and Prime 6 sheet in attach). Then we began to search the Internet for this topic. It turned out that our solution is much simpler and more beautiful than the existing solutions.
Compare, pls, with this for example https://mathworld.wolfram.com/MalfattiCircles.html .
Conclusion: first try to solve the problem yourself, and then check if there are already ready-made solutions.
Many decisions were made at a time when there were no means of numerical problem solving.
@ValeryOchkov wrote:
Congratulate me! An article was published in the journal "Mathematical Education" on this discussion of ours.
