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Question Number 39365 by behi83417@gmail.com last updated on 05/Jul/18

Commented by behi83417@gmail.com last updated on 05/Jul/18

triangle AB^▲ C,is given.  AN^▲ B,AG^▲ C,BJ^▲ C,are isoscale  triangles with:          ∠ANB=∠AGC=∠BJC=120^•   1)show that :NJ^▲ G,is equilateral.  2)find ratio of radius of incircle  of this triangle to the same of:AB^▲ C.  3)inner hexagone is:  3a)ciclic  (yes/no)  3b)regular(yes/no)  3c)additional condition(s) to   chose ”yes“ option in 3a and 3b.

$$\boldsymbol{{triangle}}\:\boldsymbol{{A}}\overset{\blacktriangle} {\boldsymbol{{B}C}},\boldsymbol{{is}}\:\boldsymbol{{given}}. \\ $$$$\boldsymbol{{A}}\overset{\blacktriangle} {\boldsymbol{{N}B}},\boldsymbol{{A}}\overset{\blacktriangle} {\boldsymbol{{G}C}},\boldsymbol{{B}}\overset{\blacktriangle} {\boldsymbol{{J}C}},\boldsymbol{{are}}\:\boldsymbol{{isoscale}} \\ $$$$\boldsymbol{{triangles}}\:\boldsymbol{{with}}: \\ $$$$\:\:\:\:\:\:\:\:\angle\boldsymbol{{ANB}}=\angle\boldsymbol{{AGC}}=\angle\boldsymbol{{BJC}}=\mathrm{120}^{\bullet} \\ $$$$\left.\mathrm{1}\right)\boldsymbol{{show}}\:\boldsymbol{{that}}\::\boldsymbol{{N}}\overset{\blacktriangle} {\boldsymbol{{J}G}},\boldsymbol{{is}}\:\boldsymbol{{equilateral}}. \\ $$$$\left.\mathrm{2}\right)\boldsymbol{{find}}\:\boldsymbol{{ratio}}\:\boldsymbol{{of}}\:\boldsymbol{{radius}}\:\boldsymbol{{of}}\:\boldsymbol{{incircle}} \\ $$$$\boldsymbol{{of}}\:\boldsymbol{{this}}\:\boldsymbol{{triangle}}\:\boldsymbol{{to}}\:\boldsymbol{{the}}\:\boldsymbol{{same}}\:\boldsymbol{{of}}:\boldsymbol{{A}}\overset{\blacktriangle} {\boldsymbol{{B}C}}. \\ $$$$\left.\mathrm{3}\right)\boldsymbol{{inner}}\:\boldsymbol{{hexagone}}\:\boldsymbol{{is}}: \\ $$$$\left.\mathrm{3}\boldsymbol{{a}}\right)\boldsymbol{{ciclic}}\:\:\left(\boldsymbol{{yes}}/\boldsymbol{{no}}\right) \\ $$$$\left.\mathrm{3}\boldsymbol{{b}}\right)\boldsymbol{{regular}}\left(\boldsymbol{{yes}}/\boldsymbol{{no}}\right) \\ $$$$\left.\mathrm{3}\boldsymbol{{c}}\right)\boldsymbol{{additional}}\:\boldsymbol{{condition}}\left(\boldsymbol{{s}}\right)\:\boldsymbol{{to}}\: \\ $$$$\boldsymbol{{chose}}\:''\boldsymbol{{yes}}``\:\boldsymbol{{option}}\:\boldsymbol{{in}}\:\mathrm{3}\boldsymbol{{a}}\:\boldsymbol{{and}}\:\mathrm{3}\boldsymbol{{b}}. \\ $$

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