Can any one tell me how to use the word hamartia in everyday writing? I have searched a lot but failed to find any sentence using this word. Any help would be appreciated.
From what I understand, "hamartia" can be used more generally to refer to a tragic or fatal flaw. There is also a medical definition that seems to be about abnormal tissue growth (though it is not a tumour). So you could probably use "hamartia" when referring to a fatal flaw, but I suspect most people who have not studied medicine or Greek tragedies will not get the reference.
In conclusion: Don't do it (unless you want to completely confuse your audience).
I think example will work (since you insisted):
The web application's hamartia was failing to sanitize input on the user sign-in form. This allowed the database to be compromised and credit card numbers to be stolen.
From Wikipedia http://en.m.wikipedia.org/wiki/Hamartia I quickly found two examples of usage:
In Greek tragedy, stories that contain a character with a hamartia often follow a similar blueprint. The hamartia, as stated, is seen as an error in judgment or unwitting mistake as applied to the actions of the hero.