# Can Isosceles Triangle Properties be used in inverse to prove?

For this question, I and my teacher had a dispute.

I argued that you can directly state that is an isosceles triangle as the angle bisector of apex and the perpendicular to the opposite side from the apex coincide. See 1 and 2 in: Therefore, I said that you can prove that 3 and 4 are true as well easily proving that S is a midpoint and allowing me to use the midpoint theorem to prove that FS//TR. But My teacher says you can't take the inverse of those properties to prove that a triangle is isosceles. According to her, you need to congruent PQS triangle and PSR triangles and proving QS=SR. Who is right?

• @JaapScherphuis But those are only true for isosceles aren't they. So doesn't 1,2 automatically imply isosceles? Jun 24, 2020 at 12:20
• (Sorry I needed to rephrase my comment so I deleted the previous one) Isosceles implies 1=2=3=4 (the four types of line coincide). The problem states only that you have 1=2. You want to use line type 3 for your proof so you need 1=3. For that to be valid you still need to show that 1=2 implies 1=3, or even that 1=2 implies isosceles. This is true, but you haven't shown it, just assumed it. Jun 24, 2020 at 12:25
• What I am asking is doesn't 1=2 automatically imply it is isosceles as 1=2 can only be if it is isoceles? Jun 24, 2020 at 12:27
• I guess the question is, how detailed does a proof need to be? How big are the steps you are allowed to take? It depends on what the axioms are and what theorems derived from those axioms are assumed to be common knowledge. In this situation it may be obvious to you that bisector=perpendicular implies isosceles. But if that hasn't been proved previously, you'll have to break it down further. You can for example apply the fact that the sum of the angles of a triangle is constant in both PQS and PSR, thereby showing that Q=R. Jun 24, 2020 at 12:43
• @JaapScherphuis That is basically what my teacher said. It isn't a theorem so can't I use the inverse of that so that those 4 implies isosceles. Jun 24, 2020 at 12:47

If $$x = 2$$ and $$y = 3$$, then $$x + y = 5$$.
So: can I now say that if $$x+y = 5$$, then $$x = 2$$ and $$y = 3$$? Of course not. The converse of a theorem (the assertion that the conclusion implies the hypotheses) is a separate statement that may or may not be true. In the event that it is is true, it requires separate proof.
You have a theorem that says "if isosceles , then conditions 1 and 2 hold"; you apparently want to claim "so if conditions 1 and 2 hold, the triangle must be isosceles." That's exactly analogous to my $$2 + 3 = 5$$ theorem.