Suddenly Urinating In Wrong Areas

Maybe it will help if 1) your mom scoops and cleans the litter box regularly at least 1x a day, 2) toilet train the dog and 3) use a better cleaning product for the carpet.
Your Yogi did not pick up or learn anything from the other pets. Unless it is a health issue (when was his last vet visit?) he is doing what is natural. He smells the urine of the other animals in the carpet and just adds his mark.

Sounds like the carpets really needs to be cleaned or even removed . If all the pets are peeing in the same spot all the time, then it is almost impossible to get the odor out totally. The carpet may have to be replaced. Go and get that plastic carpet runner and lay a section over the pee area, spike side up. Or put a litter box right in the spot where the cats are going or you can even put food or water bowls in the area. Try using the Cat Attract litter and add more litter boxes, open boxes, no hoods. Some cats want two boxes to use. One for poop and one for pee. Try adding another box near your cats other box. Your cat is only following suit with the other cat and dog in the house peeing on the carpet. I wonder if your mom's cat needs a vet check? Could be a UTI.

Get that cat vet-checked! If he's got a bladder infection or something, it'll hurt to pee and he'll think the litter box is attacking him and go somewhere else. Missing the litter box can be a red-flag to get the cat some vet care; and it's a great deal nicer to have to administer antibiotics to a cat than to have to replace soaked carpeting. Not that it hasn't gone that far already, but at least it's just the one patch.

I agree that if the carpet has been repeated peed on in that spot, you may just have to remove the carpet in that area. Usually the builder leaves some carpet scraps behind and you can just cut out the old spot and put in a new square.
And yes, get the dog potty trained and the other cat to the vet!

I don't think there is anything wrong with Yogi that would require him to see a vet. He's just putting his mark where the other animals put theirs. It's completely natural. It doesn't have anything to do with him having to pee. It's about marking so going in the litterbox wouldn't serve the purpose. It's unfortunately likely to continue for all three pets until the carpet in that spot is removed, the floor under is thoroughly cleaned and the dog stops peeing on the carpet. Dogs should not pee inside.
It sounds like the dog is not completely house broken so what you have to do is start the house training over again just like if he was a new dog or a puppy. It's the only way to do it and it's a pain but once it's done it's done and you don't have to deal with it again. Start taking him out frequently and praise him whenever he goes outside. Leave a short leash on him in the house and whenever he starts going in the spot you grab the leash and take him outside even if he's midstream. Wait until he goes outside and praise him. When I was retraining my dog after she started going inside I got up once in the middle of the night to take her out too. It was no fun at all but if I hadn't done it the floor peeing would have continued. Is the dog being walked frequently now? I'm a staunch believer in all dogs being walked at least 2-3 times a day for at least 15 minutes, preferably more. Getting him on a good, consistant walking schedule will definitely help and the dog will be happier and healthier.
If the dog is a male you can also have him wear a belly band inside so that if he pees the band will get wet which they don't like.

Putting a litterbox on the spot would be good too but since there is a dog in the picture he will just pee next to the box and the cats will do the same eventually. Until you get the carpet removed I'd clean it thoroughly with Natures Miracle and then put something on top of the spot like a box or a piece of furniture so they can't go there. Doggy will find a new spot though if he's not retrained.

It's a lot of work, I know, but it shouldn't take more than a couple of weeks and it's preferable to having carper peeing for years.

Good luck

PS. Yayi, your walking cat in your signature completely freaked me out. I just saw the tail at the bottom of my screen and thought it was a bug on my screen. Almost gave me a heart attach!