Spoiler for"...draw lines from B to X and C to X. we know these two lines are equal."
This is a flaw. The citation infers that the line BX perpendicular to AB is a right angle to CX where CX is perpendicular to AC for all triangles. While it is true that in Euclidean geometry that, except for the degenerate case where B and C are the same point, a point on the bisector of BAC will be equally distant when measured at right angles from AB and AC, the intersecting points on AB and BC are not necessarily B or C, but can be a different point or points. That is, if we select a point from AB to X at a right angle, the line segment equal in length from AC is not necessarily at point C, except when triangle BAC is an isoceles triangle. In other words, an assumption has been stated for all general triangles when it only applies to isoceles triangles where AB is equal to AC in length.
Spoiler for the reason this is wrong
... or rather, why the step made in the OP is correct is because X also lies on the perpendicular bisector of BC.