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

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: properties 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?

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  • $\begingroup$ @JaapScherphuis But those are only true for isosceles aren't they. So doesn't 1,2 automatically imply isosceles? $\endgroup$
    – VidathD
    Jun 24, 2020 at 12:20
  • $\begingroup$ (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. $\endgroup$
    – Jaap Scherphuis
    Jun 24, 2020 at 12:25
  • $\begingroup$ What I am asking is doesn't 1=2 automatically imply it is isosceles as 1=2 can only be if it is isoceles? $\endgroup$
    – VidathD
    Jun 24, 2020 at 12:27
  • $\begingroup$ 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. $\endgroup$
    – Jaap Scherphuis
    Jun 24, 2020 at 12:43
  • $\begingroup$ @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. $\endgroup$
    – VidathD
    Jun 24, 2020 at 12:47

1 Answer 1

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Here's a theorem:

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.

To answer the larger question you jumped to in the comments, "How big a leap is allowed?", the answer is "It depends on context." If Pythagoras and Aristotle are chatting, Pythagoras can probably skip a few steps, with Aristotle nodding to indicate "Sure...I can see how you'd prove that, so let's go one..." When Coxeter starts writing chapter 1 of his textbook, he has to be very thorough; by chapter 5, some proofs skip steps that he expects the reader can now fill in. But the underlying standard is always the same: if challenged, the speaker should be able to go through every step of the proof.

When you're a student in a first geometry course, and your teacher says "no, the converse of that statement isn't something we've proved; if you want to use it, you'll have to PROVE it to me," and you say "Well, isn't it completely obvious?", you've failed to meet that underlying standard. There's very little in the geometry of triangles that isn't "sorta obvious", until at some point you prove something and say, "Wait! THAT's true? That CAN'T be true! How can I have never noticed that odd thing about triangles?" ... and just about then, it gets exciting.

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