Are you looking at the current English version of the wikipedia.org article on Time Dilation? I cannot see much of what you say you are seeing.
I don't see any mention of the word "photon" in the article and I don't see any mention of the expression cΔt'. Tell me where you are getting this from.
The observer at rest does not see a hypotenuse. Where are you getting this from?
The section starts off by saying that the speed of light is constant in all reference frames. It doesn't say anything about calculating the speed of light. Where are you getting this from?
You have to be looking at something different than what I'm seeing. None of what you are talking about is in the article that I'm looking at.
You're going to have to clue me in to where you are getting all this from. I have no idea.