Engine heating up *70 kph up

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here's the story of the 36 km travel i had last saturday,

-a/c is on, at the 10 km mark i noticed temp is a little bit high above the normal range so i stopped by a gasoline station and let the engine cool down a little bit and when i opened up the radiator i can't see water inside so i poured water in it and it consumed about more than 2 liters of water, so that means water was not completely drained...

- i continued driving and on or about the 20th km mark again temperature starts to rise so i stopped again in a gasoline station and added more water with the same amount as before...

- again continued driving and the same thing happened but this time i did not return the radiator cap thinking that it would be quicker for me to pour in water even when its hot and in case the engine heats up again, and just as i thought the engine heats up again and I continued doing this routine until i went back to my starting point...

-yesterday, i poured water in the radiator and checked it this morning and lo and behold! water in the radiator is still full to the brim! so maybe i can now surmise that there's no hole in the radiator (which i have been thinking all along) or there's no problem in my cooling system, right?

so, here's the question, what could be the problem with my engine? because all along i thought my radiator was the culprit...

and oh, by the way car is toyota corolla 16v 1990 model with 4af engine