Question Number 177769 by a.lgnaoui last updated on 08/Oct/22 $${determiner}\:{les}\:{elements}\:{demandes} \\ $$$${dans}\:{le}\:{graphe} \\ $$ Commented by a.lgnaoui last updated on 08/Oct/22 Commented by a.lgnaoui last…
Question Number 177767 by mnjuly1970 last updated on 08/Oct/22 $$ \\ $$$$\:\:{in}\:{A}\overset{\Delta} {{B}C}\:{prove}\:{that}: \\ $$$$\:\:\:{r}_{{a}} \:+\:{r}_{\:{b}} \:+{r}_{\:{c}} \:=\:{r}\:+\mathrm{4}{R} \\ $$$$\:\:\:\:\:{r}_{\:{a}} \::\:{escribed}\:\:{circle}\:{radius} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{corresponding}\:{to}\:{A}. \\ $$$$\:\:\:\:{r}\::\:\:{incirle}\:{radius}\:…
Question Number 112209 by Aina Samuel Temidayo last updated on 06/Sep/20 $$\mathrm{A}\:\mathrm{triangle}\:\mathrm{ABC}\:\mathrm{has}\:\mathrm{the}\:\mathrm{following} \\ $$$$\mathrm{properties}\:\mathrm{BC}=\mathrm{1},\:\mathrm{AB}=\mathrm{BC}\:\mathrm{and}\:\mathrm{that} \\ $$$$\mathrm{the}\:\mathrm{angle}\:\mathrm{bisector}\:\mathrm{from}\:\mathrm{vertex}\:\mathrm{B}\:\mathrm{is} \\ $$$$\mathrm{also}\:\mathrm{a}\:\mathrm{median}.\:\mathrm{Find}\:\mathrm{all}\:\mathrm{possible} \\ $$$$\mathrm{triangle}\left(\mathrm{s}\right)\:\mathrm{with}\:\mathrm{its}/\mathrm{their} \\ $$$$\mathrm{side}−\mathrm{lengths}\:\mathrm{and}\:\mathrm{angles}. \\ $$ Commented…
Question Number 112199 by Aina Samuel Temidayo last updated on 06/Sep/20 $$\mathrm{Let}\:\Omega\:\mathrm{denote}\:\mathrm{the}\:\mathrm{circumcircle}\:\mathrm{of}\:\mathrm{ABC}. \\ $$$$\mathrm{The}\:\mathrm{tangent}\:\mathrm{to}\:\Omega\:\mathrm{at}\:\mathrm{A}\:\mathrm{meets}\:\mathrm{BC}\:\mathrm{at}\:\mathrm{X}. \\ $$$$\mathrm{Let}\:\mathrm{the}\:\mathrm{angle}\:\mathrm{bisectors}\:\mathrm{of}\:\angle\mathrm{AXB}\:\mathrm{meet} \\ $$$$\mathrm{AC}\:\mathrm{and}\:\mathrm{AB}\:\mathrm{at}\:\mathrm{E}\:\mathrm{and}\:\mathrm{F} \\ $$$$\mathrm{respectively}.\:\mathrm{D}\:\mathrm{is}\:\mathrm{the}\:\mathrm{foot}\:\mathrm{of}\:\mathrm{the}\:\mathrm{angle} \\ $$$$\mathrm{bisector}\:\mathrm{from}\:\angle\mathrm{BAC}\:\mathrm{on}\:\mathrm{BC}.\:\mathrm{Let}\:\mathrm{AD} \\ $$$$\mathrm{intersect}\:\mathrm{EF}\:\mathrm{at}\:\mathrm{K}\:\mathrm{and}\:\Omega\:\mathrm{again}\:\mathrm{at} \\…
Question Number 46546 by ajfour last updated on 28/Oct/18 Commented by ajfour last updated on 28/Oct/18 $${A}\:{circular}\:{ring}\:{of}\:{radius}\:{R},\:{rests} \\ $$$${against}\:{two}\:{adjacent}\:{walls}\:{and} \\ $$$${ground}.\:{If}\:{G}\left({r}\mathrm{cos}\:\alpha,\:{r}\mathrm{sin}\:\alpha,\:\mathrm{0}\right), \\ $$$${find}\:{coordinates}\:{of}\:{P}\:{and}\:{Q} \\ $$$$\left({points}\:{of}\:{contact}\:{of}\:{ring}\:{with}\right.…
Question Number 112076 by I want to learn more last updated on 06/Sep/20 Commented by Aina Samuel Temidayo last updated on 06/Sep/20 $$\mathrm{Is}\:\mathrm{OD}\:\mathrm{the}\:\mathrm{angle}\:\mathrm{bisector}\:\mathrm{of}\:\mathrm{D}? \\ $$…
Question Number 112060 by Aina Samuel Temidayo last updated on 06/Sep/20 $$\mathrm{In}\:\mathrm{a}\:\mathrm{trapezium},\:\mathrm{ABCD},\:\mathrm{with}\:\mathrm{AB} \\ $$$$\mathrm{parallel}\:\mathrm{to}\:\mathrm{CD}.\:\mathrm{If}\:\mathrm{M}\:\mathrm{is}\:\mathrm{the}\:\mathrm{midpoint}\:\mathrm{of} \\ $$$$\mathrm{line}\:\mathrm{segment}\:\mathrm{AD}\:\mathrm{and}\:\mathrm{P}\:\mathrm{is}\:\mathrm{a}\:\mathrm{point}\:\mathrm{on} \\ $$$$\mathrm{line}\:\mathrm{BC}\:\mathrm{such}\:\mathrm{that}\:\mathrm{MP}\:\mathrm{is}\:\mathrm{perpendicular} \\ $$$$\mathrm{to}\:\mathrm{BC}.\:\mathrm{Show}\:\mathrm{that},\:\mathrm{we}\:\mathrm{need}\:\mathrm{only}\:\mathrm{the} \\ $$$$\mathrm{lengths}\:\mathrm{of}\:\mathrm{line}\:\mathrm{segments}\:\mathrm{MP}\:\mathrm{and}\:\mathrm{BC} \\ $$$$\mathrm{to}\:\mathrm{calculate}\:\mathrm{the}\:\mathrm{area}\:\mathrm{ABCD}. \\…
Question Number 177596 by cherokeesay last updated on 07/Oct/22 Answered by mr W last updated on 07/Oct/22 Commented by mr W last updated on 07/Oct/22…
Question Number 112059 by Aina Samuel Temidayo last updated on 05/Sep/20 $$\mathrm{Using}\:\mathrm{the}\:\mathrm{cosine} \\ $$$$\mathrm{rule}\left(\mathrm{c}^{\mathrm{2}} =\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} −\mathrm{2abcosC}\right),\:\mathrm{prove}\:\mathrm{the} \\ $$$$\mathrm{triangle}\:\mathrm{inequality}:\:\mathrm{if}\:\mathrm{a},\mathrm{b}\:\mathrm{and}\:\mathrm{c}\:\mathrm{are} \\ $$$$\mathrm{sides}\:\mathrm{of}\:\mathrm{a}\:\mathrm{triangle}\:\mathrm{ABC},\:\mathrm{then}\:\mathrm{a}+\mathrm{b}\geqslant\mathrm{c} \\ $$$$\mathrm{and}\:\mathrm{explain}\:\mathrm{when}\:\mathrm{equality}\:\mathrm{holds}. \\ $$$$\mathrm{Further}\:\mathrm{prove}\:\mathrm{that}\:\mathrm{sin}\:\alpha\:+\:\mathrm{sin}\:\beta\:\geqslant…
Question Number 46481 by MrW3 last updated on 27/Oct/18 $${Four}\:{spheres}\:{with}\:{radii}\:{a},{b},{c}\:{and}\:{d} \\ $$$${touch}\:{each}\:{other}.\:{Find}\:{the}\:{radii}\:{of} \\ $$$${their}\:{circumscribed}\:{sphere}\:\left({R}\right)\:{and} \\ $$$${their}\:{inscribed}\:{sphere}\:\left({r}\right)\:{in}\:{terms}\:{of}\: \\ $$$${a},{b},{c}\:{and}\:{d}. \\ $$ Commented by MJS last updated…