What is the flaw in this proof that all triangles are isosceles?

First off, the other two answers here are wrong because you can prove the statement using the same flaw if P was outside the triangle. The proof using the same flaw but done in the outside of the triangle is here: [https://www.youtube.com/watch?v=Yajonhixy4g][1]

The video clearly does this proof with P on the outside. If P were on the segments of the triangle, then the proof will still hold because triangle AEP will be congruent to triangle AFP by SAA (shared sides, bisected angles, 90 degree angles). You know that P and D are the same points because the definition of P is where the perpendicular bisector meets the bisector of the opposite angle, and provided the case where P's on BC, it can only possibly be a triangle if P and D are on the same place. D is the bisector of BC, so so is P. EP = FP because we proved that AEP = AFP. BP = CP because P bisects and is between BC. Since EBP and FCP are right triangles and the Hypotenuse and one of the legs are congruent, we have triangle EBP is congruent to triangle FCP. From here, basic elementary algebra can be use to prove AB = AC.

So, what is wrong with this proof? The answer is betweenness. Every single one of these proofs rely on a certain order of points your drawing above should be:

We still have AE=AF, PE=PF, and PB=PC, and it's still true that BE=FC, but AB=AC is not true. This is because F is still between A and C, but E is not between A and B. Since E is not between A and B, SAA does not follow, and neither does SAS for this particular case.

The problem is that the perpendicular and the bisector do not intersect inside the triangle, as the picture erroneously suggests. In fact, this fallacy is a proof by contradiction that for non-isosceles triangles the intersection point $P$ is necessarily outside of the triangle.

Ravi Vakil quipped that "geometry is the art of drawing correct conclusions from incorrect pictures". The reason it works is because most geometric arguments are insensitive to many distortions, for instance the fallacy argument works as long as we move the point $P$ inside the triangle. But there are thresholds that once crossed change the picture qualitatively. This is why many arguments are split into cases. Missing a case produces fallacies, for instance forgetting that the radius of the circumscribed circle can lie outside of a triangle leads to a "proof" that in any triangle all angles are acute, and the radius is shorter than the longest side.

If the drawing is made correctly, $P$ needs to be outside, but not too outside, the triangle. If it is far outside, $E$ is on the extension of $AB$ and $F$ is on the extension of $AC$, two plus signs in the argument change to minus signs, and the argument goes through. If $P$ is not too far outside, $E$ is on the extension of $AB$, $F$ is still between $A$ and $C$ (or the other way around), one plus becomes a minus, and the argument fails.