Are domestic cats related to lions and tigers?

Yes, domestic cats are related to lions and tigers and all the other cats because they all started off at the same source. This is about evolution and taxonomy. But you have to go back about 40 million years to Miacoids (miacids) which appear to be the first cats on the planet although they did not look like modern day cats. 

Tiger and domestic cat. Collage: PoC.

It is currently believed that from that start all the cat species evolved in different places and at different times. Miacids evolved into the Proailurus (a civet/cat creature) and that creature evolved into prehistoric cats which walked on their toes like modern day cats. 

An example was a species called Pseudaelurus. This creature was the size of a modern-day cat. It evolved into two main groups one of which was Schizailurus which in turn evolved into the Felidae (the family of cats). Martelli's cat was a species which inhabited Europe about 2.4 million years ago and which is believed to be the ancestor of today's wildcat, a species of cat which is still present in Europe. Apparently the first modern day cats were cheetahs.

Cat history from the start to modern day. Cat History For Kids. Credits: Landscape of N.America 10m years ago Wikimedia Commons author Jay Matternes. In public domain USA. Fair use pleaded. Pseudaelurus in public domain USA and fair use pleaded. USA 1945 by army.arch creative commons. Snowshoe cat copyright Helmi Flick. Note: there is one small deliberate mistake in the picture. Can you spot it and leave a comment?

You have to believe in evolution to believe that domestic cats are related to lions and tigers. But science supports evolution. It is through evolution that domestic cats are related to lions and tigers. Domestic cats are very, very similar in every respect to lions and tigers. There is not much difference which is apparent because they both look like cats! There's a massive size difference but the domestic cat, in its mind, thinks just like a tiger or lion. And their behaviour is incredibly similar. That alone tells you that they are related.

If you want to know a bit more about big cat history you can click on this link. And I cover the evolution of the cat species in a simplified way on this page (click link please). The science of taxonomy which is the classification of the species is evolving itself. It was enhanced through scientific knowledge of DNA. Before DNA scientists simply observed an animal undecided whether it was similar to another animal and therefore classified it as part of a species or subspecies. That proved to be imprecise whereas DNA testing is very precise because it gets to the core of the issue.