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Category: Geometry

Question-218733

Question Number 218733 by Spillover last updated on 14/Apr/25 Commented by A5T last updated on 15/Apr/25 $$\mathrm{This}\:\mathrm{leads}\:\mathrm{to}\:\mathrm{a}\:\mathrm{contradiction}\left(\mathrm{if}\:\mathrm{the}\:\mathrm{other}\:\mathrm{chord}\right. \\ $$$$\left.\mathrm{were}\:\mathrm{also}\:\mathrm{a}\:\mathrm{diameter}\right). \\ $$ Answered by mr W…

Question-218735

Question Number 218735 by Spillover last updated on 14/Apr/25 Answered by som(math1967) last updated on 15/Apr/25 $${let}\:{rad}\:{of}\:{large}\:{circle}\:={R} \\ $$$${rad}.{of}\:{small}\:{circle}={r} \\ $$$${AD}=\mathrm{2}\sqrt{\mathrm{2}} \\ $$$$\:\:\frac{\mathrm{1}}{\mathrm{2}}×{R}×\left(\mathrm{4}+\mathrm{2}\sqrt{\mathrm{2}}\right)=\frac{\mathrm{1}}{\mathrm{2}}×\mathrm{2}×\mathrm{2} \\ $$$$\:\Rightarrow{R}=\mathrm{2}−\sqrt{\mathrm{2}}…

Question-218580

Question Number 218580 by mr W last updated on 12/Apr/25 Commented by vnm last updated on 12/Apr/25 $$ \\ $$$$\mathrm{Let}\:\mathrm{triangle}\:\mathrm{ABC}\:\mathrm{be}\:\mathrm{inscribed}\:\mathrm{in}\:\mathrm{equilateral}\:\mathrm{triangle}\:\mathrm{DEF}. \\ $$$$\mathrm{The}\:\mathrm{area}\:\mathrm{of}\:\mathrm{DEF}\:\mathrm{is}\:\mathrm{maximal}\: \\ $$$$\mathrm{if}\:\mathrm{there}\:\mathrm{exists}\:\mathrm{a}\:\mathrm{point}\:\mathrm{G}\:\mathrm{inside}\:\mathrm{ABC}\:\mathrm{such}\:\mathrm{that}\:\mathrm{GA},\:\mathrm{GB},\:\mathrm{GC}\:\mathrm{are}\:\mathrm{perpendicular}\: \\…