Semiconductor intrinsic Fermi Level? Bandgap Energy?

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kidsasd987
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http://ecee.colorado.edu/~bart/book/book/chapter2/ch2_6.htm#2_6_2
equation (2.6.24)

Equation for intrinsic Fermi level: E_i = (E_c + E_v)/2 + (1/2) k T ln(N_v/N_c), where E_i is the intrinsic Fermi level, E_c and E_v are the conduction and valence band edges, and N_v and N_c are the effective densities of states.


we have this equation for fermi level, but I wonder how this works though?
In the lecture, we assumed Ec=Eg and Ev=0 to get E_gap/2.

but I wonder how (Ec+Ev)/2 actually gives E_gap/2 instead of Ec-Ev=Eg
 

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