Equation

Prove that for any $c\in (0,1)$ the equation $$\left(1-\left(c-cx\right)^{2/3}\right)^3=(1-c^2)x^2$$ has unique real root in $(0,1)$ and find this root.

Comment. This Problem has connection with a property of the astroid - being the curve which is obtained as a contour of the set of lines ending at the two axes and with unit length.
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Property of join

 Let $X$ be path-connected and $Y$ be arbitrary topological space. Then the join $X*Y$ is simply connected.   $\textbf{Proof}.$ We use Van K...

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