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 

${𝑟}_1\cdot {𝑟}_4={𝑟}_2\cdot {𝑟}_3$.

Click on the image.