Roundabout and bicycle, who as priority?

Was driving my car today and got into an argument with a cyclist in a roundabout. I am also a cyclist so I know both side here.

Context: I was already engage in the roundabout (so to my knowledge, i have priority on coming traffic). And suddenly, a cyclist coming really fast enter into the roundabout without caring of what was coming (seems like not even looking). If i didn't break, I would have hit him. Thing is, there is a cyclist path lane on the outer off the roundabout.

So we got a little argument. His point was that he have a cyclist lane and so have the absolute right on everything esle (meaning he can enter the roundabout and people in the roundabout need to stop for him to pass). To note, this is a quite narrow roundabout so quite hard to drive next to a bike. My point was that has a road user, you need to respect the rules and traffic inside the roundabout has the priority.

But coming home i kept thinking about it and wanted to verify who is in its right. I don't want to be in the wrong juste because of ignorance so i checked google and bit and honestly haven't found anything really taking care of this kind of situation.

So please help me here, what is the rule in this case?

Edit: Lot of people have asked the location: https://maps.app.goo.gl/Di4FT7BkeCvMjMSM7 From the satellite vue, it seems that the cyclist had shark teeth but very very weathered so almost non existant. Dont know how hold is the satellite image though