I was checking the answers of the coordinate axes has minimum area question.
In short, the mentioned question asks the minimum area formed by a tangent to $y=4-x^2$ and coordinate axes. I noticed one of the answers to the question mentioned an interesting property:
Using similar triangles, it can be proven that the area of the triangle is twice the product $xy$ where $y=f(x)$.
This property seems valid for the mentioned question, as well as $y=a/x$ and several other forms that I tried. However, I cannot prove or disprove.
Can we prove or disprove the property? In what conditions is the property valid?
In another answer from your link it is proven that: $$ A=\frac{(4+x^2)^2}{4x}\ne2x(4-x^2). $$