Proof: All Triangles are Isosceles


An isosceles triangle... right?

Here is the construction of a proof that all triangles are isosceles:
  1. Draw a line bisecting the angle A.
  2. Draw a line bisecting the segment BC that is perpendicular to BC.
  3. If these two lines are parallel, then we know that we have an isosceles triangle. Assume now that they are not parallel. Then they must intersect at a point P.
  4. We may now draw lines from P to E and F that are perpendicular to AB and AC, respectively.
  5. The two triangles alpha are equal since they have equal angles share one side.
  6. Thus, PE = PF.
  7. Since DP is perpendicular to BC,the two triangles gamma must be right triangles.
  8. Since DP bisects BC, D is the midpoint of BC. Thus, the two triangles gamma share two sides and one angle and are therefore equal triangles.
  9. Thus, PB = PC.
  10. With two equal sides (PB = PC and PF = PE) and an equal angle each, the two triangles beta must therefore be equal triangles.
  11. Hence, BE + EA = CF + FA and the triangle must be isosceles, q.e.d.
Obviously something is amiss here. Can you figure it out?

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