Newbie boolean algebra question

The Majority Function is

I can see intuitively that it can be simplified to

and thus A.(B + C) + B.C

but how can I use boolean algebra to get from one to the other?

This is where I get too so far

A.B.¬C + A.¬B.C + ¬A.B.C + A.B.C

= A.B.¬C + A.¬B.C + B.C.(¬A + A)

= A.B.¬C + A.¬B.C + B.C

= A(B.¬C + ¬B.C) + B.C

but I can't figure out how to rearrange the B.¬C + ¬B.C part to get the B + C that I need. Can anyone help?