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Paul intentionally took the uncircumcised Timothy into the Temple...to challenge the legalistic ... .

Paul intentionally took the uncircumcised Timothy and circumcised him. Timothy's father was a Gentile and his mother and her mother were Jewesses, Jewels really.
Paul kept the law and did not teach the Jews in the Diaspora to forsake Moses (circumcision) and the customs, but faithfully delivered the ruling of the Jerusalem Council that the Gentiles who believe need not be circumcised and should follow the decree of the council.
Paul did not bring uncircumcised Gentiles into the Jewish area of the Temple. He was falsely accused of doing so because he had traveled to Jerusalem with Gentiles who were believers. He did not, however, take them in the areas that were forbidden to the Gentiles.
There was teach the Jews in the Diaspora to forsake Moses; only the Gentiles who believed were exempt, as the First Jerusalem Council had bound.

The entire Temple compound was considered holy, but it became increasingly more holy as one entered farther in, from east to west. King Herod had enclosed the outer court with colonnades and it was referred to as the Court of the Gentiles because the "gentiles" (non-Jews) were permitted to enter the Temple area. They could walk within in it but they were forbidden to go any further than the outer court. They were excluded from entering into any of the inner courts, and warning signs in Greek and Latin were placed that gave warning that the penalty for such trespass was death. The Romans permitted the Jewish authorities to carry out the death penalty for this offence, even if the offender were a Roman citizen. It was for this alleged crime that Paul was attacked and nearly beaten to death by an angry crowd during his last visit to Jerusalem (Acts 21:27-32).

Then came he to Derbe and Lystra: and, behold, a certain disciple was there, named Timotheus, the son of a certain woman, which was a Jewess, and believed; but his father was a Greek: Which was well reported of by the brethren that were at Lystra and Iconium. Him would Paul have to go forth with him; and took and circumcised him because of the Jews which were in those quarters: for they knew all that his father was a Greek.

Paul, an apostle of Jesus Christ by the will of God, according to the promise of life which is in Christ Jesus, To Timothy, my dearly beloved son: Grace, mercy, and peace, from God the Father and Christ Jesus our Lord. I thank God, whom I serve from my forefathers with pure conscience, that without ceasing I have remembrance of thee in my prayers night and day; Greatly desiring to see thee, being mindful of thy tears, that I may be filled with joy; When I call to remembrance the unfeigned faith that is in thee, which dwelt first in thy grandmother Lois, and thy mother Eunice; and I am persuaded that in thee also. Wherefore I put thee in remembrance that thou stir up the gift of God, which is in thee by the putting on of my hands. For God hath not given us the spirit of fear; but of power, and of love, and of a sound mind.

And the day following Paul went in with us unto James; and all the elders were present. And when he had saluted them, he declared particularly what things God had wrought among the Gentiles by his ministry. And when they heard it, they glorified the Lord, and said unto him, Thou seest, brother, how many thousands of Jews there are which believe; and they are all zealous of the law: And they are informed of thee, that thou teachest all the Jews which are among the Gentiles to forsake Moses, saying that they ought not to circumcise their children, neither to walk after the customs. What is it therefore? the multitude must needs come together: for they will hear that thou art come. Do therefore this that we say to thee: We have four men which have a vow on them; Them take, and purify thyself with them, and be at charges with them, that they may shave their heads: and all may know that those things, whereof they were informed concerning thee, are nothing; but that thou thyself also walkest orderly, and keepest the law. As touching the Gentiles which believe, we have written and concluded that they observe no such thing, save only that they keep themselves from things offered to idols, and from blood, and from strangled, and from fornication. Then Paul took the men, and the next day purifying himself with them entered into the temple, to signify the accomplishment of the days of purifcation, until that an offering should be offered for every one of them. And when the seven days were almost ended, the Jews which were of Asia, when they saw him in the temple, stirred up all the people, and laid hands on him, Crying out, Men of Israel, help: This is the man, that teacheth all men every where against the people, and the law, and this place: and further brought Greeks also into the temple, and hath polluted this holy place. (For they had seen before with him in the city Trophimus an Ephesian, whom they supposed that Paul had brought into the temple.)
Acts, Catholic chapter sixteen, Protestant verses one to three,
Second Timothy, Catholic chapter one, Protestant verses one to seven,
Acts, Catholic chapter twenty one, Protestant verses eighteen to twenty nine,
as authorized, but not authored, by King James