"Mistake on the page (an aside, Bertrand's Paradox)"
Dear Alexander, I found a mistake in your thoughts in the subject Bertrand's Paradox, an Aside.(http://www.cut-the-knot.org/bertrand.shtml) You wrote that the triangle divides the small circle into three EQUAL parts, but that is wrong, since the triangle divides the small cirlce in the ratio 1/5, 3/5, 1/5. (Easy prove calculating the sizes of the different areas with the formula to calculate sction areas!!) Therefore your constructions don't prove the calculated first solution!! Regards Max
1. "RE: Mistake on the page (an aside, Bertrand's Paradox)"
In response to message #0
I appreciate your being on alert.
However, the small circle - not the disk, but the circle, the curve - is divided into 3 equal parts, exactly as the big circle is divided by the equilateral triangle. The two circles are homothetic from A.\
Also, the remark is not supposed to prove anything, but to establish the plausibility of the argument.
2. "RE: Mistake on the page (an aside, Bertrand's Paradox)"
In response to message #1
Ok. But then it wil be easier to understand for the reader if you write that it divides the curve of the circle into three equal parts and not the cirle itself. Thank you for the prompt answer. Regards Max
3. "RE: Mistake on the page (an aside, Bertrand's Paradox)"
In response to message #2
Ah, do not know about that. The circle terminology is strange. The word circle is used to describe both 1- abd 2-dimensional objects. The latter sometimes is referred to as a "disk", and the former as "circumference". Although, usually the word "circumeference" is reserved for the length of the border of the circle (or the disk depending on the usage.) But this is really rare.
4. "RE: Mistake on the page (an aside, Bertrand's Paradox)"
In response to message #2
I agree with this to some degree. I also though something was fishy at first! :) My mother tongue (Slovene) knows two (or more) words that are used and a reader knows right away what is meant, the curve or also the inside. It is not definite in English, I agree. The Mathematica app though has two graphical primitives: Circle and Disk.
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