Question Number 16074 by Tinkutara last updated on 17/Jun/17 $$\mathrm{Let}\:{K},\:{L},\:{M}\:\mathrm{and}\:{N}\:\mathrm{be}\:\mathrm{the}\:\mathrm{midpoints}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{sides}\:{AB},\:{BC},\:{CD}\:\mathrm{and}\:{DA}, \\ $$$$\mathrm{respectively},\:\mathrm{of}\:\mathrm{a}\:\mathrm{cyclic}\:\mathrm{quadrilateral} \\ $$$${ABCD}.\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{orthocenters} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{triangles}\:{AKN},\:{BKL},\:{CLM}\:\mathrm{and} \\ $$$${DMN}\:\mathrm{are}\:\mathrm{the}\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{a} \\ $$$$\mathrm{parallelogram}. \\ $$ Terms…
Question Number 147144 by gsk2684 last updated on 18/Jul/21 $${prove}\:{that}\: \\ $$$${equation}\:{of}\:{a}\:{circle}\:{passing}\:{through} \\ $$$${the}\:{points}\:{of}\:{intersection}\:{of}\:{a}\:{circle} \\ $$$${S}=\mathrm{0}\:{and}\:{a}\:{line}\:{L}=\mathrm{0}\:{can}\:{be}\:{taken}\:{as} \\ $$$${S}+\lambda{L}=\mathrm{0}\:{where}\:\lambda\:{is}\:{a}\:{parameter} \\ $$ Answered by mr W last…
Question Number 16075 by Tinkutara last updated on 17/Jun/17 $$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{perpendiculars} \\ $$$$\mathrm{dropped}\:\mathrm{from}\:\mathrm{the}\:\mathrm{midpoints}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sides} \\ $$$$\mathrm{of}\:\mathrm{a}\:\mathrm{cyclic}\:\mathrm{quadrilateral}\:\mathrm{to}\:\mathrm{the}\:\mathrm{opposite} \\ $$$$\mathrm{sides}\:\mathrm{are}\:\mathrm{concurrent}. \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 16071 by Tinkutara last updated on 21/Jun/17 $$\mathrm{Let}\:{A}',\:{B}'\:\mathrm{and}\:{C}'\:\mathrm{be}\:\mathrm{points}\:\mathrm{on}\:\mathrm{the}\:\mathrm{sides} \\ $$$${BC},\:{CA}\:\mathrm{and}\:{AB}\:\mathrm{of}\:\mathrm{the}\:\mathrm{triangle}\:{ABC}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{circumcircles}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{triangles}\:{AB}'{C}',\:{BA}'{C}'\:\mathrm{and}\:{CA}'{B}' \\ $$$$\mathrm{have}\:\mathrm{a}\:\mathrm{common}\:\mathrm{point}.\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{property}\:\mathrm{holds}\:\mathrm{even}\:\mathrm{if}\:\mathrm{the}\:\mathrm{points}\:{A}', \\ $$$${B}'\:\mathrm{and}\:{C}'\:\mathrm{are}\:\mathrm{collinear}. \\ $$ Answered…
Question Number 16072 by Tinkutara last updated on 17/Jun/17 $$\mathrm{Let}\:{ABCD}\:\mathrm{be}\:\mathrm{a}\:\mathrm{convex}\:\mathrm{quadrilateral}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{orthocenters}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{triangles}\:{ABC},\:{BCD},\:{CDA}\:\mathrm{and}\:{DAB} \\ $$$$\mathrm{are}\:\mathrm{the}\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{a}\:\mathrm{quadrilateral} \\ $$$$\mathrm{congruent}\:\mathrm{to}\:{ABCD}\:\mathrm{and}\:\mathrm{prove}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{centroids}\:\mathrm{of}\:\mathrm{the}\:\mathrm{same}\:\mathrm{triangles}\:\mathrm{are}\:\mathrm{the} \\ $$$$\mathrm{vertices}\:\mathrm{of}\:\mathrm{a}\:\mathrm{cyclic}\:\mathrm{quadrilateral}. \\ $$ Commented…
Question Number 16068 by Tinkutara last updated on 21/Jun/17 $$\mathrm{Let}\:{ABCD}\:\mathrm{be}\:\mathrm{a}\:\mathrm{convex}\:\mathrm{quadrilateral} \\ $$$$\mathrm{and}\:\mathrm{let}\:\mathrm{E}\:\mathrm{and}\:\mathrm{F}\:\mathrm{be}\:\mathrm{the}\:\mathrm{points}\:\mathrm{of} \\ $$$$\mathrm{intersections}\:\mathrm{of}\:\mathrm{the}\:\mathrm{lines}\:{AB},\:{CD}\:\mathrm{and} \\ $$$${AD},\:{BC},\:\mathrm{respectively}.\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{midpoints}\:\mathrm{of}\:\mathrm{the}\:\mathrm{segments}\:{AC},\:{BD}, \\ $$$$\mathrm{and}\:{EF}\:\mathrm{are}\:\mathrm{collinear}. \\ $$ Answered by ajfour…
Question Number 16069 by Tinkutara last updated on 21/Jun/17 $$\mathrm{In}\:\mathrm{the}\:\mathrm{interior}\:\mathrm{of}\:\mathrm{a}\:\mathrm{quadrilateral} \\ $$$${ABCD},\:\mathrm{consider}\:\mathrm{a}\:\mathrm{variable}\:\mathrm{point}\:{P}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{if}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{distances}\:\mathrm{from} \\ $$$${P}\:\mathrm{to}\:\mathrm{the}\:\mathrm{sides}\:\mathrm{is}\:\mathrm{constant},\:\mathrm{then}\:{ABCD} \\ $$$$\mathrm{is}\:\mathrm{a}\:\mathrm{parallelogram}. \\ $$ Terms of Service Privacy Policy…
Question Number 16070 by Tinkutara last updated on 21/Jun/17 $$\mathrm{Let}\:{ABCD}\:\mathrm{be}\:\mathrm{a}\:\mathrm{convex}\:\mathrm{quadrilateral}. \\ $$$$\mathrm{Prove}\:\mathrm{that} \\ $$$${AB}.{CD}\:+\:{AD}.{BC}\:=\:{AC}.{BD} \\ $$$$\mathrm{if}\:\mathrm{and}\:\mathrm{only}\:\mathrm{if}\:{ABCD}\:\mathrm{is}\:\mathrm{cyclic}\:\left(\mathrm{Ptolemy}'\mathrm{s}\right. \\ $$$$\left.\mathrm{theorem}\right). \\ $$ Terms of Service Privacy Policy…
Question Number 16065 by Tinkutara last updated on 21/Jun/17 $$\mathrm{Let}\:{ABCD}\:\mathrm{be}\:\mathrm{a}\:\mathrm{convex}\:\mathrm{quadrilateral}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{points}\:{M}\:\mathrm{in}\:\mathrm{its}\:\mathrm{interior} \\ $$$$\mathrm{such}\:\mathrm{that}\:\left[{MAB}\right]\:=\:\mathrm{2}\left[{MCD}\right]. \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 16066 by Tinkutara last updated on 21/Jun/17 $$\mathrm{Let}\:{ABCD}\:\mathrm{be}\:\mathrm{a}\:\mathrm{convex}\:\mathrm{quadrilateral} \\ $$$$\mathrm{and}\:\mathrm{let}\:{k}\:>\:\mathrm{0}\:\mathrm{be}\:\mathrm{a}\:\mathrm{real}\:\mathrm{number}.\:\mathrm{Find} \\ $$$$\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{points}\:{M}\:\mathrm{in}\:\mathrm{its}\:\mathrm{interior} \\ $$$$\mathrm{such}\:\mathrm{that} \\ $$$$\left[{MAB}\right]\:+\:\mathrm{2}\left[{MCD}\right]\:=\:{k}. \\ $$ Answered by mrW1 last updated…