Imagine a right angle composed of one leg of height one unit and the other leg of length x units. Now attach another leg of one unit at an angle of 89 degrees at the end of leg x. Finally, complete the quadrilateral by connecting the two legs of unit one. Next find the perpendicular bisectors and their intersection inside the quadrilateral and label that point A. From point A, connect each of the four corners of the quadrilateral, forming a total of six triangles. With the current construction it can be easily proven, with the help of side-angle-side and side-side-side theorems, that the angle which had been constructed to be acute should hypothetically be a right angle.
So what went wrong? Unless there's faulty reasoning behind the trigonometric theorems, this supposed proof seems to be justified. However, the problem within this proof does not lie within the theorems used to prove it, but rather within the original assumption of the intersection being within the quadrilateral. See the figure below. The perpendicular bisectors will never intersect inside the quadrilateral and thus all the operations mentioned above are simply for a fictional scenario.
