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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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