New answers tagged euclidean-geometry
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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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