Question Number 82416 by TawaTawa last updated on 21/Feb/20 Commented by TawaTawa last updated on 21/Feb/20 $$\mathrm{Please}\:\mathrm{help}. \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 16882 by RasheedSoomro last updated on 27/Jun/17 $$\:^{\bullet} \mathrm{In}\:\mathrm{plane}\:\mathrm{parallel}\:\mathrm{lines}\:\mathrm{are}\:\mathrm{the}\:\mathrm{lines} \\ $$$$\mathrm{which}\:\mathrm{don}'\mathrm{t}\:\mathrm{meet}\:\mathrm{each}\:\mathrm{other}. \\ $$$$ \\ $$$$\:^{\bullet} \mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{condition}\:\mathrm{in}\:\mathrm{space} \\ $$$$\mathrm{that}\:\mathrm{two}\:\mathrm{lines}\:\mathrm{be}\:\mathrm{parallel}? \\ $$ Commented by prakash…
Question Number 16879 by Tinkutara last updated on 27/Jun/17 $$\mathrm{Let}\:{ABCD}\:\mathrm{be}\:\mathrm{a}\:\mathrm{parallelogram}.\:\mathrm{The} \\ $$$$\mathrm{points}\:{M},\:{N}\:\mathrm{and}\:{P}\:\mathrm{are}\:\mathrm{chosen}\:\mathrm{on}\:\mathrm{the} \\ $$$$\mathrm{segments}\:{BD},\:{BC}\:\mathrm{and}\:{CD}, \\ $$$$\mathrm{respectively},\:\mathrm{so}\:\mathrm{that}\:{CNMP}\:\mathrm{is}\:\mathrm{a} \\ $$$$\mathrm{parallelogram}.\:\mathrm{Let}\:{E}\:=\:{AN}\:\cap\:{BD}\:\mathrm{and} \\ $$$${F}\:=\:{AP}\:\cap\:{BD}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\left[{AEF}\right]\:=\:\left[{DFP}\right]\:+\:\left[{BEN}\right]. \\ $$ Terms…
Question Number 16877 by Tinkutara last updated on 27/Jun/17 $$\mathrm{From}\:\mathrm{a}\:\mathrm{point}\:\mathrm{on}\:\mathrm{the}\:\mathrm{circumcircle}\:\mathrm{of}\:\mathrm{an} \\ $$$$\mathrm{equilateral}\:\mathrm{triangle}\:{ABC}\:\mathrm{parallels}\:\mathrm{to} \\ $$$$\mathrm{the}\:\mathrm{sides}\:{BC},\:{CA}\:\mathrm{and}\:{AB}\:\mathrm{are}\:\mathrm{drawn}, \\ $$$$\mathrm{intersecting}\:\mathrm{the}\:\mathrm{sides}\:{CA},\:{AB}\:\mathrm{and}\:{BC} \\ $$$$\mathrm{at}\:\mathrm{the}\:\mathrm{points}\:{M},\:{N},\:{P},\:\mathrm{respectively}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{points}\:{M},\:{N}\:\mathrm{and}\:{P}\:\mathrm{are} \\ $$$$\mathrm{collinear}. \\ $$ Terms…
Question Number 16878 by Tinkutara last updated on 27/Jun/17 $$\mathrm{Let}\:{P}\:\mathrm{be}\:\mathrm{a}\:\mathrm{point}\:\mathrm{on}\:\mathrm{the}\:\mathrm{circumcircle}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{equilateral}\:\mathrm{triangle}\:{ABC}.\:\mathrm{Prove} \\ $$$$\mathrm{that}\:\mathrm{the}\:\mathrm{projections}\:\mathrm{of}\:\mathrm{any}\:\mathrm{point}\:{Q} \\ $$$$\mathrm{onto}\:\mathrm{the}\:\mathrm{lines}\:{PA},\:{PB}\:\mathrm{and}\:{PC}\:\mathrm{are}\:\mathrm{the} \\ $$$$\mathrm{vertices}\:\mathrm{of}\:\mathrm{an}\:\mathrm{equilateral}\:\mathrm{triangle}. \\ $$ Commented by prakash jain last…
Question Number 16875 by Tinkutara last updated on 27/Jun/17 $$\mathrm{Let}\:{P}_{\mathrm{1}} ,\:{P}_{\mathrm{2}} ,\:…,\:{P}_{{n}} \:\mathrm{be}\:\mathrm{a}\:\mathrm{convex}\:\mathrm{polygon} \\ $$$$\mathrm{with}\:\mathrm{the}\:\mathrm{following}\:\mathrm{property}\::\:\mathrm{for}\:\mathrm{any} \\ $$$$\mathrm{two}\:\mathrm{vertices}\:{P}_{{i}} \:\mathrm{and}\:{P}_{{j}} ,\:\mathrm{there}\:\mathrm{exists}\:\mathrm{a} \\ $$$$\mathrm{vertex}\:{P}_{{k}} \:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{segment}\:{P}_{{i}} {P}_{{j}} \\ $$$$\mathrm{is}\:\mathrm{seen}\:\mathrm{from}\:{P}_{{k}}…
Question Number 16873 by Tinkutara last updated on 27/Jun/17 $$\mathrm{Let}\:{I}\:\mathrm{be}\:\mathrm{the}\:\mathrm{incenter}\:\mathrm{of}\:\Delta{ABC}.\:\mathrm{It}\:\mathrm{is} \\ $$$$\mathrm{known}\:\mathrm{that}\:\mathrm{for}\:\mathrm{every}\:\mathrm{point}\:{M}\:\in\:\left({AB}\right), \\ $$$$\mathrm{one}\:\mathrm{can}\:\mathrm{find}\:\mathrm{the}\:\mathrm{points}\:{N}\:\in\:\left({BC}\right)\:\mathrm{and} \\ $$$${P}\:\in\:\left({AC}\right)\:\mathrm{such}\:\mathrm{that}\:{I}\:\mathrm{is}\:\mathrm{the}\:\mathrm{centroid}\:\mathrm{of} \\ $$$$\Delta{MNP}.\:\mathrm{Prove}\:\mathrm{that}\:{ABC}\:\mathrm{is}\:\mathrm{an} \\ $$$$\mathrm{equilateral}\:\mathrm{triangle}. \\ $$ Terms of Service…
Question Number 16874 by Tinkutara last updated on 27/Jun/17 $$\mathrm{Let}\:{ABC}\:\mathrm{be}\:\mathrm{an}\:\mathrm{acute}\:\mathrm{triangle}.\:\mathrm{The} \\ $$$$\mathrm{interior}\:\mathrm{bisectors}\:\mathrm{of}\:\mathrm{the}\:\mathrm{angles}\:\angle{B}\:\mathrm{and} \\ $$$$\angle{C}\:\mathrm{meet}\:\mathrm{the}\:\mathrm{opposite}\:\mathrm{sides}\:\mathrm{at}\:\mathrm{the} \\ $$$$\mathrm{points}\:{L}\:\mathrm{and}\:{M},\:\mathrm{respectively}.\:\mathrm{Prove} \\ $$$$\mathrm{that}\:\mathrm{there}\:\mathrm{exists}\:\mathrm{a}\:\mathrm{point}\:{K}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{interior}\:\mathrm{of}\:\mathrm{the}\:\mathrm{side}\:{BC}\:\mathrm{such}\:\mathrm{that} \\ $$$$\Delta{KLM}\:\mathrm{is}\:\mathrm{equilateral}\:\mathrm{if}\:\mathrm{and}\:\mathrm{only}\:\mathrm{if} \\ $$$$\angle{A}\:=\:\mathrm{60}°. \\…
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Question Number 82378 by ajfour last updated on 20/Feb/20 Commented by ajfour last updated on 20/Feb/20 $$\mathrm{If}\:\mathrm{all}\:\mathrm{5}\:\mathrm{regions}\:\mathrm{have}\:\mathrm{equal}\:\mathrm{areas}, \\ $$$$\mathrm{find}\:\mathrm{sides}\:\mathrm{of}\:\mathrm{rectangle},\:\mathrm{given} \\ $$$$\mathrm{radius}=\mathrm{1}. \\ $$ Answered by…