Category: Geometry

Question Number 16739 by Tinkutara last updated on 26/Jun/17 $$\mathrm{Let}M\mathrm{be}\mathrm{a}\mathrm{point}\mathrm{in}\mathrm{the}\mathrm{interior}\mathrm{of}\mathrm{the}$$$$\mathrm{equilateral}\mathrm{triangle}ABC\mathrm{and}\mathrm{let}{A}^{\prime },$$$$B^{\prime }\mathrm{and}{C}^{\prime }\mathrm{be}\mathrm{its}\mathrm{projections}\mathrm{onto}\mathrm{the}$$$$\mathrm{sides}BC,CA\mathrm{and}AB,\mathrm{respectively}.$$$$\mathrm{Prove}\mathrm{that}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{lengths}\mathrm{of}\mathrm{the}$$$$\mathrm{inradii}\mathrm{of}\mathrm{triangles}{MAC}^{\prime },{MBA}^{\prime }\mathrm{and}$$$${MCB}^{\prime }\mathrm{equals}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{lengths}\mathrm{of}\mathrm{the}$$$$\mathrm{inradii}\:\mathrm{of}\:\mathrm{trianges}\:{MAB}',\:{MBC}'\:\mathrm{and} \…

Question Number 16736 by Tinkutara last updated on 26/Jun/17 $$\mathrm{The}\mathrm{side}\mathrm{lengths}\mathrm{of}\mathrm{an}\mathrm{equiangular}$$$$\mathrm{octagon}\mathrm{are}\mathrm{rational}\mathrm{numbers}.\mathrm{Prove}$$$$\mathrm{that}\mathrm{the}\mathrm{octagon}\mathrm{has}\mathrm{a}\mathrm{symmetry}$$$$\mathrm{center}.$$ Terms of Service Privacy Policy Contact: info@tinkutara.com

Question Number 16734 by Tinkutara last updated on 26/Jun/17 $$\mathrm{An}\mathrm{equiangular}\mathrm{polygon}\mathrm{with}\mathrm{an}\mathrm{odd}$$$$\mathrm{number}\mathrm{of}\mathrm{sides}\mathrm{is}\mathrm{inscribed}\mathrm{in}\mathrm{a}\mathrm{circle}.$$$$\mathrm{Prove}\mathrm{that}\mathrm{the}\mathrm{polygon}\mathrm{is}\mathrm{regular}.$$ Terms of Service Privacy Policy Contact: info@tinkutara.com

Question Number 16665 by b.e.h.i.8.3.4.1.7@gmail.com last updated on 25/Jun/17 Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 25/Jun/17 $$ABCDE,isregular{.}^{\prime }{F}^{\prime }isapointon$$$$cicumcircle.$$$$1)provethat:$$$$FD+FA+FC=FE+FB$$$$\left.\mathrm{2}\right){if}:\:{DC}=\mathrm{1},{FA}=\sqrt{\mathrm{5}},{find}\:{all}\:{segments}…

Question Number 16613 by ajfour last updated on 24/Jun/17 Commented by ajfour last updated on 24/Jun/17 $$\mathrm{From}\mathrm{the}\mathrm{endpoint}\mathrm{of}\mathrm{a}\mathrm{diameter}$$$$\mathrm{of}\mathrm{a}\mathrm{sphere}\mathrm{draw}\mathrm{a}\mathrm{chord}\mathrm{so}\mathrm{that}$$$$\mathrm{the}\mathrm{surface}\mathrm{generated}\mathrm{by}\mathrm{a}\mathrm{rotation}$$$$\mathrm{about}\mathrm{this}\mathrm{diameter}\mathrm{divides}\mathrm{the}$$$$\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sphere}\:\mathrm{into}\:\mathrm{two}…