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There is a famous theorem in elementary geometry:

Theorem. An isosceles triangle with a $60^\circ$ angle is equilateral.

Two cases of this theorem are depicted below. I consider any (or both) of these cases.

Two cases of the same theorem

The standard proof relies on two key facts:

  • The base angles of any isosceles triangle are equal in measure.
  • The sum of interior angles of a triangle equals a straight angle ($180^\circ$).

I'd like to find a proof that does not rely on the second fact. I would like it to involve some of these ones and/or corollaries from these:

  • Basic axioms (namely, Pogorelov axiomatics).
  • If two angles are supplementary, then they add up to a straight angle.
  • Triangle congruence theorems (SAS, ASA, SSS).
  • In an isosceles triangle, the median, altitude, and angle bisector from the vertex angle all coincide.
  • For the triangle to be isosceles, it's necessary and sufficient for it to have two equal interior angles.

Is there a proof or a way to prove the impossibility of such proof? What's the minimum amount of dependent theorems known to prove that?

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    $\begingroup$ Are you assuming the 60 degree angle is not between the two assumed equal sides? $\endgroup$ Commented Nov 22, 2025 at 20:37
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    $\begingroup$ @coffeemath any of those, or both of those. $\endgroup$ Commented Nov 22, 2025 at 20:55

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Within the bounds you specified, you cannot prove this fact. Why? Because every theorem you named is provable without the so-called parallel postulate, and thus is valid in both hyperbolic geometry and standard plane geometry.

However, in hyperbolic geometry, there are isosceles triangles with 60 degree apex angles and less than 60 degree base angles.

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  • $\begingroup$ So without the parallel postulate, it's unprovable? Good insight. $\endgroup$ Commented Nov 22, 2025 at 20:57
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    $\begingroup$ In hyperbolic geometry there are also isosceles triangles with 60 degree base angles and less than 60 degree apex angles. The essential issue is whether the three angles of a triangle add up to 180 degrees or not. $\endgroup$ Commented Nov 22, 2025 at 21:20
  • $\begingroup$ Well, without the parallel postulate the sum of the angles of a triangle is not 180. That should be a basic fact. $\endgroup$ Commented Nov 22, 2025 at 22:11
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    $\begingroup$ the fourth edition of Marvin Greenberg's Euclidean and Non-Euclidean Geometries has a short teacher's manual that includes 70 properties equivalent to the parallel postulate. I have it around here somewhere... I see, 77 statements, pages 30-34. $\endgroup$ Commented Nov 23, 2025 at 2:09

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