Question 5
Let $(O)$ be the circle with diameter $AB$, and let $C$ be any point on segment $AB$. Construct circles $(O_1),(O_2)$ with diameters $AC,CB$. Let $M$ lie on $(O)$, distinct from $A,B$, and let $ME,MF$ be tangents to $(O_1),(O_2)$, touching at $E,F$.
a) Prove $\left(\dfrac{ME}{MA}\right)^2+\left(\dfrac{MF}{MB}\right)^2=1$.
b) Find $C$ such that $ME=\dfrac12MA$.
a) Prove $\left(\dfrac{ME}{MA}\right)^2+\left(\dfrac{MF}{MB}\right)^2=1$.
b) Find $C$ such that $ME=\dfrac12MA$.