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Category: Geometry

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 19104 by Tinkutara last updated on 04/Aug/17 LetABCbeanacuteangledtrianglewithACBCandletObethecircumcenterandFbethefootofaltitudethroughC.Further,letXandYbethefeetofperpendicularsdroppedfromAandBrespectivelyto(theextensionof)CO.ThelineFOintersects$$\mathrm{the}\:\mathrm{circumcircle}\:\mathrm{of}\:\Delta\mathrm{FXY},\:\mathrm{second}\:\mathrm{time} \

Let-PQRS-be-a-rectangle-such-that-PQ-a-and-QR-b-Suppose-r-1-is-the-radius-of-the-circle-passing-through-P-and-Q-and-touching-RS-and-r-2-is-the-radius-of-the-circle-passing-through-Q-and-R-and-t

Question Number 18967 by Tinkutara last updated on 02/Aug/17 LetPQRSbearectanglesuchthatPQ=aandQR=b.Supposer1istheradiusofthecirclepassingthroughPandQandtouchingRSandr2istheradiusofthecirclepassingthroughQandRandtouchingPS.Showthat:$$\mathrm{5}\left({a}\:+\:{b}\right)\:\leqslant\:\mathrm{8}\left(\mathrm{r}_{\mathrm{1}} \:+\:\mathrm{r}_{\mathrm{2}}…