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Category: Geometry

Question-151973

Question Number 151973 by ajfour last updated on 24/Aug/21 Commented by ajfour last updated on 24/Aug/21 $${OB}={OC}=\sqrt{{OA}}\:\:;\:{OM}=\mathrm{1} \\ $$$${yellow}\:{area}=\mathrm{1}/\mathrm{4}\:,\:{then}\:{find} \\ $$$${OB}={OC}\:={x}\:\:\:{or}\:\:{OA}={x}^{\mathrm{2}} . \\ $$ Answered…

In-a-rectangle-ABCD-E-is-the-midpoint-of-AB-F-is-a-point-on-AC-such-that-BF-is-perpendicular-to-AC-and-FE-perpendicular-to-BD-Suppose-BC-8-3-Find-AB-

Question Number 20599 by Tinkutara last updated on 28/Aug/17 $$\mathrm{In}\:\mathrm{a}\:\mathrm{rectangle}\:{ABCD},\:{E}\:\mathrm{is}\:\mathrm{the}\:\mathrm{midpoint} \\ $$$$\mathrm{of}\:{AB};\:{F}\:\mathrm{is}\:\mathrm{a}\:\mathrm{point}\:\mathrm{on}\:{AC}\:\mathrm{such}\:\mathrm{that}\:{BF} \\ $$$$\mathrm{is}\:\mathrm{perpendicular}\:\mathrm{to}\:{AC};\:\mathrm{and}\:{FE} \\ $$$$\mathrm{perpendicular}\:\mathrm{to}\:{BD}.\:\mathrm{Suppose}\:{BC}\:=\:\mathrm{8}\sqrt{\mathrm{3}}. \\ $$$$\mathrm{Find}\:{AB}. \\ $$ Answered by ajfour last updated…

Let-ABC-be-an-acute-angled-triangle-with-AC-BC-and-let-O-be-the-circumcenter-and-F-be-the-foot-of-altitude-through-C-Further-let-X-and-Y-be-the-feet-of-perpendiculars-dropped-from-A-and-B-respecti

Question Number 20545 by Tinkutara last updated on 23/Sep/17 $$\mathrm{Let}\:\mathrm{ABC}\:\mathrm{be}\:\mathrm{an}\:\mathrm{acute}-\mathrm{angled}\:\mathrm{triangle} \\ $$$$\mathrm{with}\:\mathrm{AC}\:\neq\:\mathrm{BC}\:\mathrm{and}\:\mathrm{let}\:\mathrm{O}\:\mathrm{be}\:\mathrm{the} \\ $$$$\mathrm{circumcenter}\:\mathrm{and}\:\mathrm{F}\:\mathrm{be}\:\mathrm{the}\:\mathrm{foot}\:\mathrm{of} \\ $$$$\mathrm{altitude}\:\mathrm{through}\:\mathrm{C}.\:\mathrm{Further},\:\mathrm{let}\:\mathrm{X}\:\mathrm{and} \\ $$$$\mathrm{Y}\:\mathrm{be}\:\mathrm{the}\:\mathrm{feet}\:\mathrm{of}\:\mathrm{perpendiculars}\:\mathrm{dropped} \\ $$$$\mathrm{from}\:\mathrm{A}\:\mathrm{and}\:\mathrm{B}\:\mathrm{respectively}\:\mathrm{to}\:\left(\mathrm{the}\right. \\ $$$$\left.\mathrm{extension}\:\mathrm{of}\right)\:\mathrm{CO}.\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{FY}\:\bot\:\mathrm{CA} \\ $$$$\mathrm{using}\:\mathrm{that}\:\angle\mathrm{CFY}\:=\:\angle\mathrm{CBY}\:=\:\angle\mathrm{CAF}. \\…