  We construct the three-variable p-adic triple product L-functions attached to
Hida families of ellptic newforms and prove the explicit interpolation formulae
at all critical specializations by establishing explicit Ichino's formulae for
the trilinear period integrals of automorphic forms. Our formulae perfectly fit
the conjectural shape of p-adic L-functions predicted by Coates and
Perrin-Riou. As an application, we prove the factorization of certain
unbalanced p-adic triple product L-functions into a product of anticyclotomic
p-adic L-functions for modular forms. By this factorization, we give a new
construction of the anticyclotomic p-adic L-functions for elliptic curves in
the definite case via the diagonal cycle Euler system \'a la Darmon and Rotger
and obtain a Greenberg-Stevens style proof of anticyclotomic exceptional zero
conjecture for elliptic curves due to Bertolini and Darmon.
