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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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