Question Number 81267 by behi83417@gmail.com last updated on 10/Feb/20
$$\boldsymbol{\mathrm{p}},\mathrm{is}\:\mathrm{a}\:\mathrm{point},\boldsymbol{\mathrm{inside}}\:,\boldsymbol{\mathrm{onside}}\:,\boldsymbol{\mathrm{outside}}\:\mathrm{of} \\ $$$$\mathrm{equilateral}\:\mathrm{triangle}.\mathrm{find}\:\mathrm{side}\:\mathrm{of}\:\mathrm{triangle} \\ $$$$\mathrm{if}\:\mathrm{distance}\:\mathrm{of}\::\boldsymbol{\mathrm{p}}\:\mathrm{from}\:\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{triangle} \\ $$$$\mathrm{be}\:\mathrm{equail}\:\mathrm{to}:\:\mathrm{5},\mathrm{7},\mathrm{11}. \\ $$$$\left(\mathrm{study}\:\mathrm{each}\:\mathrm{conditions}\:\mathrm{separately}\right). \\ $$$$\mathrm{find}\:\mathrm{side}\:\mathrm{of}\:\mathrm{ABC}\:\mathrm{and}\:\mathrm{p}_{\mathrm{1}} \mathrm{p}_{\mathrm{2}} \mathrm{p}_{\mathrm{3}} \:\mathrm{in}\:\mathrm{a}\: \\ $$$$\mathrm{special}\:\mathrm{case}\:\mathrm{that}:\begin{cases}{\mathrm{Ap}_{\mathrm{1}} =\mathrm{11}}\\{\mathrm{Bp}_{\mathrm{2}} =\mathrm{5}}\\{\mathrm{Cp}_{\mathrm{3}} =\mathrm{7}}\end{cases} \\ $$
Commented by behi83417@gmail.com last updated on 10/Feb/20
Commented by mr W last updated on 10/Feb/20
$${sir}:\:{it}'{s}\:{not}\:{clear}\:{what}\:{is}\:{meant}. \\ $$$${if}\:{the}\:{distance}\:{from}\:{a}\:{point}\:{P},\:{which} \\ $$$${can}\:{lie}\:{inside},\:{onside}\:{or}\:{outside}\:{of}\:{the} \\ $$$${triangle},\:{to}\:{the}\:{vertexes}\:{of}\:{the}\:{triangle} \\ $$$${is}\:\mathrm{5},\mathrm{7},\mathrm{11}\:{respectively},\:{then}\:{the}\:{side} \\ $$$${length}\:{of}\:{the}\:{triangle}\:{is}\:{also}\:{fixed}, \\ $$$${generally}\:{there}\:{are}\:{two}\:{values}. \\ $$$$ \\ $$$${just}\:{to}\:{have}\:{AP}_{\mathrm{1}} =\mathrm{11},\:{BP}_{\mathrm{2}} =\mathrm{5}\:{and}\: \\ $$$${CP}_{\mathrm{3}} =\mathrm{7},\:{ABC}\:{and}\:{P}_{\mathrm{1}} {P}_{\mathrm{2}} {P}_{\mathrm{3}} \:{could}\:{be}\:{of} \\ $$$${infinitely}\:{many}\:{sizes},\:{i}.{e}.\:{you}\:{can} \\ $$$${not}\:{determine}\:{triangles}\:\:{ABC}\:{and}\:{P}_{\mathrm{1}} {P}_{\mathrm{2}} {P}_{\mathrm{3}} \\ $$$${using}\:{only}\:{this}\:{condition}. \\ $$
Commented by behi83417@gmail.com last updated on 11/Feb/20
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{dear}\:\mathrm{master}.\mathrm{there}\:\mathrm{is}\:\mathrm{3}\:\mathrm{separate} \\ $$$$\mathrm{questions}: \\ $$$$\mathrm{1}.\:\mathrm{p},\mathrm{is}\:\mathrm{inside}\:\mathrm{of}\:\mathrm{triangle}\:\mathrm{and}\:\mathrm{its}\:\mathrm{distances} \\ $$$$\mathrm{from}\:\mathrm{vertices}\:\mathrm{are}\::\mathrm{5},\mathrm{7},\mathrm{11}.\:\mathrm{and}\:\mathrm{so}\:\mathrm{on}\:\mathrm{for} \\ $$$$\mathrm{q}#\mathrm{2}:\boldsymbol{\mathrm{onside}}\:\mathrm{and}\:\mathrm{q}#\mathrm{3}:\boldsymbol{\mathrm{outsie}}. \\ $$$$\mathrm{each}\:\mathrm{colore}\:\mathrm{is}\:\mathrm{for}\:\mathrm{one}\:\mathrm{question}. \\ $$
Commented by behi83417@gmail.com last updated on 11/Feb/20
Commented by mr W last updated on 11/Feb/20
$${i}\:{see}. \\ $$$${but}\:{if}\:{the}\:{three}\:{distances}\:{are}\:{given}, \\ $$$${we}\:{can}\:{uniquely}\:{determine}\:{the} \\ $$$${corresponding}\:{equilateral}\:{triangle}. \\ $$$${we}\:{can}\:{then}\:{check}\:{if}\:{the}\:{point}\:{is}\: \\ $$$${inside},\:{onside}\:{or}\:{outside}\:{the}\:{triangle}. \\ $$$${but}\:{this}\:{is}\:{already}\:{solved}\:{in}\:{an}\:{earlier} \\ $$$${post}\:{of}\:{you}. \\ $$