Four Circles Tangent to an Ellipse

Problem 

An ellipse is inscribed in an angle. Four circles $𝑜_1, 𝑜_2, 𝑜_3, 𝑜_4$ with radii $𝑟_1, 𝑟_2, 𝑟_3, 𝑟_4$, respectively, are also inscribed in the given angle and are tangent to the ellipse, as show in following figure. 

Prove that 

$𝑟_1 ⋅𝑟_4 = 𝑟_2 ⋅𝑟_3$.

Click on the image.