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Category: Geometry

Question-46864

Question Number 46864 by ajfour last updated on 01/Nov/18 Commented by ajfour last updated on 02/Nov/18 $${If}\:{a}\:{sphere}\:{of}\:{radius}\:\boldsymbol{{R}}\:{touches} \\ $$$${the}\:{walls}\:{at}\:{A},\:{B},\:{C}\:;\:{find}\:\boldsymbol{{a}},\boldsymbol{{b}},\boldsymbol{{c}} \\ $$$${in}\:{terms}\:{of}\:\alpha,\beta,\gamma\:{and}\:{R}. \\ $$ Commented by…

A-triangle-ABC-has-the-following-properties-BC-1-AB-AC-and-that-the-angle-bisector-from-vertex-B-is-also-a-median-Find-all-possible-triangle-s-with-its-their-side-lengths-and-angles-

Question Number 112318 by Aina Samuel Temidayo last updated on 07/Sep/20 $$\mathrm{A}\:\mathrm{triangle}\:\mathrm{ABC}\:\mathrm{has}\:\mathrm{the}\:\mathrm{following} \\ $$$$\mathrm{properties}\:\mathrm{BC}=\mathrm{1},\:\boldsymbol{\mathrm{AB}}=\boldsymbol{\mathrm{AC}}\:\mathrm{and}\:\mathrm{that} \\ $$$$\mathrm{the}\:\mathrm{angle}\:\mathrm{bisector}\:\mathrm{from}\:\mathrm{vertex}\:\mathrm{B}\:\mathrm{is} \\ $$$$\mathrm{also}\:\mathrm{a}\:\mathrm{median}.\:\mathrm{Find}\:\mathrm{all}\:\mathrm{possible} \\ $$$$\mathrm{triangle}\left(\mathrm{s}\right)\:\mathrm{with}\:\mathrm{its}/\mathrm{their} \\ $$$$\mathrm{side}−\mathrm{lengths}\:\mathrm{and}\:\mathrm{angles}. \\ $$ Commented…

Let-denote-the-circumcircle-of-ABC-The-tangent-to-at-A-meets-BC-at-X-Let-the-angle-bisectors-of-AXB-meet-AC-and-AB-at-E-and-F-respectively-D-is-the-foot-of-the-angle-bisector-from-BAC-on-BC-

Question Number 112266 by Aina Samuel Temidayo last updated on 07/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} \\…

in-AB-C-prove-that-r-a-r-b-r-c-r-4R-r-a-escribed-circle-radius-corresponding-to-A-r-incirle-radius-R-circumcircle-radius-

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}\:…

A-triangle-ABC-has-the-following-properties-BC-1-AB-BC-and-that-the-angle-bisector-from-vertex-B-is-also-a-median-Find-all-possible-triangle-s-with-its-their-side-lengths-and-angles-

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…