First off, lets make an observation. The following is true (if you don't believe me, I'll prove it later):
\$ A + B = AB + \bar AB + A \bar B \$
Now, by the definition of XOR we have:
\$ A \oplus B = \bar AB + A \bar B \$
Combining the two expressions we get:
\$ A + B = AB + A \oplus B \$
With that, lets begin:
AB+AC+BC = AB + C(A+B) // Factor out C
= AB + C(AB + A⨁B) // Substitute the above expression
= AB + CAB + C(A⨁B) // Factor out AB
= AB + C(A⨁B) // Invoke the absorption rule to get rid of CAB
Voila, that's what we were looking for. I don't claim that is the best way to do it, but it seems to work well enough.
Now, I'm going to show \$ A + B = AB + \bar AB + A \bar B \$ using sweet, pure Boolean logic.
\$ A+B = A(\bar B + B) + B(\bar A + A) = AB + \bar AB + A \bar B \$