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We construct the three-variable $p$-adic triple product $L$-functions attached to Hida families of elliptic 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 obtain a construction of the square root 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.