Let us suppose that God possesses all perfections. His love is perfect. His knowledge is perfect. He creates all things from nothing. He is a personal God, not a Deist clockwork.
Now God’s knowledge is perfect. But what does God know? All of creation, of course. But first and foremost, God knows himself.
Since whatever God does is done perfectly, He also knows himself perfectly.
What does this mean? It means that there is nothing lacking in God’s knowledge of Himself. There is nothing whatever lacking in his knowledge of himself. His knowledge is a perfect mirror of himself, possessing every perfection of God, including even personhood. (Otherwise, his knowing would be imperfect.) St. John calls this the divine logos, or the Word: God speaking his knowledge of himself is the Divine Word.
In the beginning was the Word,
And the Word was with God
And the Word was God.
It is not a creation, God creating a new being. The Word is rather his own being perfectly expressed—begotten, not made. Of one essence. One nature. One God, expressed in personhood as Father and Son.
I am not going to continue with the theology of the Holy Spirit; others have done it better than me. None of it is contradictory. It is no more a contradiction for God to be three persons in one nature than it is for you and I to be one person in one nature. (If I am one person in one nature, shouldn’t I be two rather than one? No. If God is three persons in one nature, shouldn’t he be three beings rather than one? No.)
While the nature of God as trinity is not something that we could have arrived at without divine revelation, there is nothing contradictory about it. Nobody is saying that three natures = one nature, or that three persons = one person. We affirm one nature, one being, one essence in God, but a trinity of persons. Persons are not distinct entities from their nature; they are only distinct in personhood.