Question Number 16738 by Tinkutara last updated on 26/Jun/17 $$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{segments}\:\mathrm{joining}\:\mathrm{the} \\ $$$$\mathrm{midpoints}\:\mathrm{of}\:\mathrm{the}\:\mathrm{opposite}\:\mathrm{sides}\:\mathrm{of}\:\mathrm{an} \\ $$$$\mathrm{equiangular}\:\mathrm{hexagon}\:\mathrm{are}\:\mathrm{concurrent}. \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 16739 by Tinkutara last updated on 26/Jun/17 $$\mathrm{Let}\:{M}\:\mathrm{be}\:\mathrm{a}\:\mathrm{point}\:\mathrm{in}\:\mathrm{the}\:\mathrm{interior}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equilateral}\:\mathrm{triangle}\:{ABC}\:\mathrm{and}\:\mathrm{let}\:{A}', \\ $$$${B}'\:\mathrm{and}\:{C}'\:\mathrm{be}\:\mathrm{its}\:\mathrm{projections}\:\mathrm{onto}\:\mathrm{the} \\ $$$$\mathrm{sides}\:{BC},\:{CA}\:\mathrm{and}\:{AB},\:\mathrm{respectively}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{lengths}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{inradii}\:\mathrm{of}\:\mathrm{triangles}\:{MAC}',\:{MBA}'\:\mathrm{and} \\ $$$${MCB}'\:\mathrm{equals}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{lengths}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{inradii}\:\mathrm{of}\:\mathrm{trianges}\:{MAB}',\:{MBC}'\:\mathrm{and} \\…
Question Number 16737 by Tinkutara last updated on 26/Jun/17 $$\mathrm{A}\:\mathrm{convex}\:\mathrm{hexagon}\:\mathrm{is}\:\mathrm{given}\:\mathrm{in}\:\mathrm{which} \\ $$$$\mathrm{any}\:\mathrm{two}\:\mathrm{opposite}\:\mathrm{sides}\:\mathrm{have}\:\mathrm{the} \\ $$$$\mathrm{following}\:\mathrm{property}:\:\mathrm{the}\:\mathrm{distance} \\ $$$$\mathrm{between}\:\mathrm{their}\:\mathrm{midpoints}\:\mathrm{is}\:\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\:\mathrm{times}\:\mathrm{the} \\ $$$$\mathrm{sum}\:\mathrm{of}\:\mathrm{their}\:\mathrm{lengths}.\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{hexagon}\:\mathrm{is}\:\mathrm{equiangular}. \\ $$ Terms of Service…
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 82273 by TawaTawa last updated on 19/Feb/20 Commented by mr W last updated on 19/Feb/20 $$\frac{{h}}{\mathrm{12}}=\frac{\mathrm{6}}{\mathrm{8}} \\ $$$$\Rightarrow{h}=\mathrm{9} \\ $$ Commented by TawaTawa…
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 16735 by Tinkutara last updated on 26/Jun/17 $$\mathrm{Let}\:{a}_{\mathrm{1}} ,\:{a}_{\mathrm{2}} ,\:…,\:{a}_{{n}} \:\mathrm{be}\:\mathrm{the}\:\mathrm{side}\:\mathrm{lengths}\:\mathrm{of}\: \\ $$$$\mathrm{an}\:\mathrm{equiangular}\:\mathrm{polygon}.\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{if} \\ $$$${a}_{\mathrm{1}} \:\geqslant\:{a}_{\mathrm{2}} \:\geqslant\:…\:\geqslant\:{a}_{{n}} ,\:\mathrm{then}\:\mathrm{the}\:\mathrm{polygon}\:\mathrm{is} \\ $$$$\mathrm{regular}. \\ $$ Terms…
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 $${ABCDE},{is}\:{regular}.'{F}'\:{is}\:{a}\:{point}\:{on} \\ $$$${cicumcircle}. \\ $$$$\left.\mathrm{1}\right){prove}\:{that}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{FD}+{FA}+{FC}={FE}+{FB} \\ $$$$\left.\mathrm{2}\right){if}:\:{DC}=\mathrm{1},{FA}=\sqrt{\mathrm{5}},{find}\:{all}\:{segments}…
Question Number 16641 by Tinkutara last updated on 24/Jun/17 $$\mathrm{Prove}\:\mathrm{that}\:{p}\:\mathrm{is}\:\mathrm{a}\:\mathrm{prime}\:\mathrm{number}\:\mathrm{if}\:\mathrm{and} \\ $$$$\mathrm{only}\:\mathrm{if}\:\mathrm{every}\:\mathrm{equiangular}\:\mathrm{polygon}\:\mathrm{with} \\ $$$${p}\:\mathrm{sides}\:\mathrm{of}\:\mathrm{rational}\:\mathrm{lengths}\:\mathrm{is}\:\mathrm{regular}. \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
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}…