Is the Christian concept of the Trinity influenced by Philo's philosophy? Philo, the Hellenistic Jewish philosopher who lived at the same time as Jesus (albeit in Egypt) and during the time of the early church, taught a trinity very similar to what the Scriptures teach concerning the Trinity. 

Philo taught that there were three divine Beings, the first, God, the second, the Logos, and the third, the Divine or Holy Spirit, and that God created the Logos and the Divine Spirit from his own "essence". 

He said that God created through the Logos, and he said that the Logos was the "first-born", the Governor and Administrator of All Things, a High Priest, and that Adam was made in his image. He also said that the Logos would be the Messiah, sent as a son to do the will of his father, who is God. 

Philo taught that the Divine or Holy Spirit would possess people, be breathed on people, and would lead them to seek God, and that it proceeds from God. 

Christian doctrine teaches all of these things, except that the Logos and the Holy Spirit were created by God, but instead teaches that they themselves are God, although subordinate to the Father. Philo, however, was born significantly before the advent of Christianity, and died just around the time that Saint Paul would begin writing his letters; in other words, he came before Christianity. 

How can it be shown that Christian theology was not influenced by Philo's philosophy, or that they at least came to the same conclusions using that same source, namely, the Old Testament?
Answer

Original question: Was the Christian concept of the trinity inspired by Philo's philosophy? 

No, the Christian concept of the Trinity is not inspired by Philo's philosophy. It seems most probable that Philo based some of his writings on the Trinity from his knowledge of Jesus and Holy Spirit in the events of the Gospels, some writings as a philosophical expansion on the previous renderings of the Logos, and some writings based on the Old Testament, as the word Logos in the Septuagint appears quite often. See this link. Additionally, you asked 


  How can it be shown that Christian theology was not influenced by Philo's philosophy?


"Influenced by" is quite distinct that "inspired by," unless you meant these words to be synonymous. "Inspired by" generally entails being the source of something, vs an "influence" just affects something else in any way, shape, form, or extent. I would say that Christian theology is/was influenced by Philo, probably even a much smaller extent compared to Arian. So the exact answer would be I don't think it can be shown, because it seems probable that Christian theology was influenced by Philo, though not inspired by. 

Discussion

The exact answer to your question depends on what exactly you mean by the "Christian concept of the Trinity." If you are referring to the creeds formulated at Nicaea, then it is probable that those were affected by Philo's heterodox teachings. If you are referring to the doctrine of the Trinity as believed by ante-Nicene church fathers, then it is also probable that Philo's writings affected them, since ante-Nicene includes all of 1st century up until 325 AD. If you are referring to the doctrine of the Trinity itself, then no...see below. 

An important element of this question hinges on the dating of Philo's works, since he lived 20 BC to AD 50. If the Christian concept of the Trinity is inspired by Philo's writings, then his relevant writings should be prior to when the Christian concept of the Trinity was established. I have unsuccessfully attempted to find any date given to Philo's writings, of which the ones including the Trinity and the Logos being the most relevant and central to this question. Philo spent a large portion of his writings and his life on Genesis and the Pentateuch rather than the Trinity. 

The concept of the logos is familiar to Greek philosophy prior to Jesus' time and starting in the 5th century BC with Heraclitus, though with somewhat of an inconsistent and transient usage. Aristotle used logos in combination with ethos and pathos in his writings on persuasion. Read more here. Philo's discussion of the Logos could be a purely philosophically-based expansion on the previous philosophical/theological renderings of Logos, as the Logos had already had theological implications at least with the Stoics in the 4th century BC.

The New Testament writings were most likely all written after the death of Philo in AD 50; however, the events of the Gospels all happened by AD 30.  The earliest church father I am aware of besides the New Testament writers, Ignatius of Antioch, was born in AD 35. I doubt he would have been able to write a treatise on Jesus as the Logos before Philo did. I don't know if we can know how the apostles applied the term Logos to Jesus before the Gospel accounts. In Eusebius' Ecclesiastical History, Peter and Philo supposedly met, which is relevant. It is also possible that Philo was familiar with the events occurring with Jesus, including the Holy Spirit, Pentecost, "breathing" the Holy Spirit on the disciples, etc., which explains some of Philo's writings on the Holy Spirit (pneumatology) and the Trinity. He probably (possibly?) would have been familiar with what the Christians taught and believed about Jesus, so that most likely would have affected his writings. 

You could prove me false by demonstrating that Philo's writings on the Trinity pre-date the events of the Gospels (before ~AD 25). It is difficult to otherwise separate what Philo knew of Christian beliefs at the time of his writings. Jesus was born around 4-6 BC, so his ministry started around AD ~25-27, so if Philo wrote extensively on the Trinity (especially and specifically pneumatology) before the events described in the Gospels, then I would be wrong. Otherwise, I would think it is the Gospel events (not the books) that are Philo's primary basis for his doctrine.  To have such sophisticated work on the Trinity would be extremely surprising based on the Old Testament alone. 

Conclusion

In short, this is a great and difficult question; I don't think we have the data to come to a solid conclusion on this. And this was a terrible answer but just wanted to give you some information for your consideration in finding an answer to this question. Summary: I think Philo based his writings on the Logos from the Old Testament and philosophy, but his writings on pneumatology the events contained in the Gospels that he would have heard about.

Edit: This link (also linked in the Answer section) helped me see you could have some substantive writings on the Logos from the Old Testament alone, especially with a philosophical background because Logos appears a decent amount in the Septuagint, the Greek translation of the Hebrew Scriptures (Old Testament).

Therefore, it seems that Philo bases his ideas of the Logos on philosophy + Old Testament considering that he does quote it, see this answer, but the ideas of pneumatology on Gospel events since those seem to be very specifically from Jesus' ministry and teachings, and probably some overlap between the two. Jesus often referred to the Old Testament to reveal the Trinity, so Philo could have been aware of that.    

P.S. if someone can give me dates for any/all of Philo's writings, that would be wonderful.