Of course this proof must not be true, my task is to find the place where this proof falls apart, after lotta tries, I think i've finally found out what's wrong, take a look at this image:
(Sorry the quality sucks cause i'm still a beginner in GeoGebra)
In this image, I've drawn a right angled triangle with sides 3-4-5, I've followed what the proof says
The proof says that AF = AE and FC = BE, this is case 4 cause they meet outside the triangle so
AF - FC = AE - BE
AF - FC = AC
AE - BE is not equal to AB
That's where the proof falls apart
The proof tried to use an unproven statement that says that:
"In case 4, AF - FC = AC and AE - BE = AB"
and this may not necessarily be the case
Did I nail it? or did I make a mistake?
"As for case 3, I think it is always isosceles", is this statement correct?
Actually cases 2 and 3 can't occur