Answer:
[tex]\dfrac{2ab}{a+b}.[/tex]
Step-by-step explanation:
Consider the trapezoid ABCD. In this trapezoid BC=a and AD=b.
Since triangles BOC and AOD are somilar, then
[tex]\dfrac{AO}{OC}=\dfrac{DO}{OB}=\dfrac{AD}{BC}=\dfrac{b}{a}.[/tex]
Triangles OAE and CAB are similar, then
[tex]\dfrac{EO}{BC}=\dfar{AO}{AC}=\dfrac{b}{a+b}.[/tex]
This means that
[tex]EO=\dfrac{ba}{b+a}.[/tex]
Similarly, from similar triangles FDO and CDB:
[tex]OF=\dfrac{ba}{b+a}.[/tex]
Thus,
[tex]EF=\dfrac{2ab}{a+b}.[/tex]
