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As $BI$ is mean proportional to $BD$ and $BE$. \begin{array}{l} \Rightarrow \frac{B D}{B I}=\frac{B I}{B E} \\ \Rightarrow \frac{B D}{B I} \times B D=\frac{B I}{B E} \times B D \\ \Rightarrow \frac{B D^{2}}{B I}=\frac{B I \times B D}{B E} \\ \Rightarrow \quad \frac{B D^{2}}{B I^{2}}=\frac{B D}{B E} \end{array} $Q.E.D$


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Wells' book is available for download on PdfDrive. Puzzle #38 appears in the section of problems attributed to Abū al-Wafā and is preceeded by the comment that "he is best known for his study of geometrical dissections and of constructions with a rusty compass, meaning a compass which is so stiff that it can be used with only one opening". In the ...


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Denote $\beta=\angle MBC,\;\alpha=\angle DMN,\;\gamma=\angle CDH$. We have: $$\tan\beta=1/2,\quad\cos\alpha=-\sin\beta=-1/\sqrt{5},\quad\sin\alpha=\cos\beta=2/\sqrt{5}.$$ Take $DM=1$, then $MN=\sqrt{5}-2$, and let $x=DN$. By the rule of cosines, $$x^2=1^2+(\sqrt{5}-2)^2+2(\sqrt{5}-2)/\sqrt{5}=12-24/\sqrt{5}.$$ Then by the rule of sines, $$\sin\gamma=\frac{(\...


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Wikipedia: Languages using duodecimal number systems are uncommon. Languages in the Nigerian Middle Belt such as Janji, Gbiri-Niragu (Gure-Kahugu), Piti, and the Nimbia dialect of Gwandara and the Chepang language of Nepal are known to use duodecimal numerals. Also: It is thought that Nimbia, which is isolated from the rest of Gwandara, acquired its ...


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The first link says that communication complexity studies the amount of communication required to solve a problem when the problem is distributed between two or more parties. Whilst the second link says that Kolmogorov complexity of an object, such as a piece of text, is the length of the shortest computer program... that produces the object as output. It ...


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