Charles Link's latest activity

You are right. There is one more approach which is based on the figure from the post #39 and it is simpler than the one I have provided...

Looking back to the other posts, I see @Gavran also used Ptolemy's theorem previously in post 43, but the post 52 method gets the...

@hutchphd Here is the solution: ## (ac)(bd)=2x^2+(11)(2) ## from Ptolemy ## (ac)^2=4x^2-2^2 ## from Pythagoras and Thales ##...

Please feel free. I have had an exhausting week and seem disinclined to do anything (feeling each of my 73 yrs I guess.....ah, well. I...

@hutchphd a super result. :) With a couple algebraic steps one gets ## 3x^2=11^2+2^2+(2)(11) ##. I'll let you write it out, but if you...

There is also a lovely theorem by Ptolemy that deals with cyclic quadrilaterals. I think that will work......not an intuitive result...

Here is an aside item, and in a way a little simpler. The problem of finding integer solutions for ## x^2=(a^2+ab+b^2)/3 ## reminds me...

and a follow-on to the above: The case of ## m=1 ## gives the point ## (1,1) ## for ## (x,y) ##. It may be worth mentioning that I...

As others have noted, if the four sides of that quadrilateral have lengths a, x, b, and the diameter 2x, then 3x^2 = a^2+ab+b^2. As to...

I think I figured out now what @daverusin did in post 41. One begins with ## x^2+xy+y^2-3=0 ##, and to find the rational solutions...

I found the following video that might explain some of post 41. It seems to be a specialized topic, and not one that is found in the...

Could you elaborate on this a little? (post 41). My mathematics with calculus, etc. is reasonably advanced, but this looks like it is...

One more approach based on https://en.wikipedia.org/wiki/Cyclic_quadrilateral and the cosine theorem. ## \angle...

Good catch. To eliminate solution triples that are integer multiples of other triples I must verify that either ##b## or ##x## (or both)...

@kuruman Very good. :) Excellent derivation. Between you and @renormalize of post 32, we have a simpler expression for ## R=x ## than...