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Question Number 38849 by behi83417@gmail.com last updated on 30/Jun/18

Commented by behi83417@gmail.com last updated on 30/Jun/18

raw idea to mr Ajfour! a^2 =x^2 +t^2 ,z=c.cos𝛉 b^2 =x^2 +y^2 ,x=1βˆ’c.cos𝛉 c^2 =y^2 +z^2 ,t=1βˆ’c.sin𝛉 d^2 =z^2 +t^2 ,y=c.sin𝛉 βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’ a^2 +c^2 =b^2 +d^2

$$\:\boldsymbol{{raw}}\:\boldsymbol{{idea}}\:\boldsymbol{{to}}\:\boldsymbol{{mr}}\:\boldsymbol{{Ajfour}}! \\ $$$$\boldsymbol{{a}}^{\mathrm{2}} =\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{t}}^{\mathrm{2}} \:\:,\boldsymbol{{z}}=\boldsymbol{{c}}.\boldsymbol{{cos}\theta} \\ $$$$\boldsymbol{{b}}^{\mathrm{2}} =\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} \:\:\:,\boldsymbol{{x}}=\mathrm{1}βˆ’\boldsymbol{{c}}.\boldsymbol{{cos}\theta} \\ $$$$\boldsymbol{{c}}^{\mathrm{2}} =\boldsymbol{{y}}^{\mathrm{2}} +\boldsymbol{{z}}^{\mathrm{2}} \:\:\:\:,\boldsymbol{{t}}=\mathrm{1}βˆ’\boldsymbol{{c}}.\boldsymbol{{sin}\theta} \\ $$$$\boldsymbol{{d}}^{\mathrm{2}} =\boldsymbol{{z}}^{\mathrm{2}} +\boldsymbol{{t}}^{\mathrm{2}} \:\:\:\:\:,\boldsymbol{{y}}=\boldsymbol{{c}}.\boldsymbol{{sin}\theta} \\ $$$$βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{a}}^{\mathrm{2}} +\boldsymbol{{c}}^{\mathrm{2}} =\boldsymbol{{b}}^{\mathrm{2}} +\boldsymbol{{d}}^{\mathrm{2}} \\ $$

Answered by ajfour last updated on 30/Jun/18

ultimately all of these leads to (if ccot ΞΈ =x ) x^2 (x^2 +c^2 )=(x+c)^2 .

$${ultimately}\:{all}\:{of}\:{these}\:{leads}\:{to} \\ $$$$\left({if}\:\:\boldsymbol{{c}}\mathrm{cot}\:\theta\:={x}\:\right) \\ $$$$\:\:\:\:\boldsymbol{{x}}^{\mathrm{2}} \left(\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{c}}^{\mathrm{2}} \right)=\left(\boldsymbol{{x}}+\boldsymbol{{c}}\right)^{\mathrm{2}} \:. \\ $$

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