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Question Number 81267 by behi83417@gmail.com last updated on 10/Feb/20

p,is a point,inside ,onside ,outside of  equilateral triangle.find side of triangle  if distance of :p from  vertices of triangle  be equail to: 5,7,11.  (study each conditions separately).  find side of ABC and p_1 p_2 p_3  in a   special case that: { ((Ap_1 =11)),((Bp_2 =5)),((Cp_3 =7)) :}

$$\boldsymbol{\mathrm{p}},\mathrm{is}\:\mathrm{a}\:\mathrm{point},\boldsymbol{\mathrm{inside}}\:,\boldsymbol{\mathrm{onside}}\:,\boldsymbol{\mathrm{outside}}\:\mathrm{of} \\ $$$$\mathrm{equilateral}\:\mathrm{triangle}.\mathrm{find}\:\mathrm{side}\:\mathrm{of}\:\mathrm{triangle} \\ $$$$\mathrm{if}\:\mathrm{distance}\:\mathrm{of}\::\boldsymbol{\mathrm{p}}\:\mathrm{from}\:\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{triangle} \\ $$$$\mathrm{be}\:\mathrm{equail}\:\mathrm{to}:\:\mathrm{5},\mathrm{7},\mathrm{11}. \\ $$$$\left(\mathrm{study}\:\mathrm{each}\:\mathrm{conditions}\:\mathrm{separately}\right). \\ $$$$\mathrm{find}\:\mathrm{side}\:\mathrm{of}\:\mathrm{ABC}\:\mathrm{and}\:\mathrm{p}_{\mathrm{1}} \mathrm{p}_{\mathrm{2}} \mathrm{p}_{\mathrm{3}} \:\mathrm{in}\:\mathrm{a}\: \\ $$$$\mathrm{special}\:\mathrm{case}\:\mathrm{that}:\begin{cases}{\mathrm{Ap}_{\mathrm{1}} =\mathrm{11}}\\{\mathrm{Bp}_{\mathrm{2}} =\mathrm{5}}\\{\mathrm{Cp}_{\mathrm{3}} =\mathrm{7}}\end{cases} \\ $$

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 5,7,11 respectively, then the side  length of the triangle is also fixed,  generally there are two values.    just to have AP_1 =11, BP_2 =5 and   CP_3 =7, ABC and P_1 P_2 P_3  could be of  infinitely many sizes, i.e. you can  not determine triangles  ABC and P_1 P_2 P_3   using only this condition.

$${sir}:\:{it}'{s}\:{not}\:{clear}\:{what}\:{is}\:{meant}. \\ $$$${if}\:{the}\:{distance}\:{from}\:{a}\:{point}\:{P},\:{which} \\ $$$${can}\:{lie}\:{inside},\:{onside}\:{or}\:{outside}\:{of}\:{the} \\ $$$${triangle},\:{to}\:{the}\:{vertexes}\:{of}\:{the}\:{triangle} \\ $$$${is}\:\mathrm{5},\mathrm{7},\mathrm{11}\:{respectively},\:{then}\:{the}\:{side} \\ $$$${length}\:{of}\:{the}\:{triangle}\:{is}\:{also}\:{fixed}, \\ $$$${generally}\:{there}\:{are}\:{two}\:{values}. \\ $$$$ \\ $$$${just}\:{to}\:{have}\:{AP}_{\mathrm{1}} =\mathrm{11},\:{BP}_{\mathrm{2}} =\mathrm{5}\:{and}\: \\ $$$${CP}_{\mathrm{3}} =\mathrm{7},\:{ABC}\:{and}\:{P}_{\mathrm{1}} {P}_{\mathrm{2}} {P}_{\mathrm{3}} \:{could}\:{be}\:{of} \\ $$$${infinitely}\:{many}\:{sizes},\:{i}.{e}.\:{you}\:{can} \\ $$$${not}\:{determine}\:{triangles}\:\:{ABC}\:{and}\:{P}_{\mathrm{1}} {P}_{\mathrm{2}} {P}_{\mathrm{3}} \\ $$$${using}\:{only}\:{this}\:{condition}. \\ $$

Commented by behi83417@gmail.com last updated on 11/Feb/20

thank you dear master.there is 3 separate  questions:  1. p,is inside of triangle and its distances  from vertices are :5,7,11. and so on for  q#2:onside and q#3:outsie.  each colore is for one question.

$$\mathrm{thank}\:\mathrm{you}\:\mathrm{dear}\:\mathrm{master}.\mathrm{there}\:\mathrm{is}\:\mathrm{3}\:\mathrm{separate} \\ $$$$\mathrm{questions}: \\ $$$$\mathrm{1}.\:\mathrm{p},\mathrm{is}\:\mathrm{inside}\:\mathrm{of}\:\mathrm{triangle}\:\mathrm{and}\:\mathrm{its}\:\mathrm{distances} \\ $$$$\mathrm{from}\:\mathrm{vertices}\:\mathrm{are}\::\mathrm{5},\mathrm{7},\mathrm{11}.\:\mathrm{and}\:\mathrm{so}\:\mathrm{on}\:\mathrm{for} \\ $$$$\mathrm{q}#\mathrm{2}:\boldsymbol{\mathrm{onside}}\:\mathrm{and}\:\mathrm{q}#\mathrm{3}:\boldsymbol{\mathrm{outsie}}. \\ $$$$\mathrm{each}\:\mathrm{colore}\:\mathrm{is}\:\mathrm{for}\:\mathrm{one}\:\mathrm{question}. \\ $$

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.

$${i}\:{see}. \\ $$$${but}\:{if}\:{the}\:{three}\:{distances}\:{are}\:{given}, \\ $$$${we}\:{can}\:{uniquely}\:{determine}\:{the} \\ $$$${corresponding}\:{equilateral}\:{triangle}. \\ $$$${we}\:{can}\:{then}\:{check}\:{if}\:{the}\:{point}\:{is}\: \\ $$$${inside},\:{onside}\:{or}\:{outside}\:{the}\:{triangle}. \\ $$$${but}\:{this}\:{is}\:{already}\:{solved}\:{in}\:{an}\:{earlier} \\ $$$${post}\:{of}\:{you}. \\ $$

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