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Given two distinct parallel lines and two distinct fixed points on one of the lines and a point that can vary on the other line. Then the areas of all the triangles formed by those 3 points are all equal. – Does this theorem have a name? I am not asking for an explanation of the mathematics, but simply an elevator-pitch name, something that rolls off the tongue more smoothly than "Do you know the area = one half base times height theorem?"

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    area = 1/2 base times height? – coffeemath Aug 15 '18 at 20:13
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    I.37 – Micah Aug 15 '18 at 20:13
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    @Micah +1, hope you don't mind that I added your reference at the end of my answer. – dxiv Aug 15 '18 at 20:21
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    @EulerSpoiler Please recall that if the OP is solved you can evaluate to accept an answer among the given, more details here https://meta.stackexchange.com/questions/5234/how-does-accepting-an-answer-work – user Sep 06 '18 at 23:57
  • @gimusi: It appears that the answer to my question is 'no'. What would you suggest I do? –  Sep 07 '18 at 12:58
  • @EulerSpoiler it seems you have received two good answer, I wanted remind you that you can accept an aswer if it is a good answer to your OP. Otherwise of course you are not forced to accept any of them. Bye – user Sep 07 '18 at 15:26
  • @gimusi: I'm surmising that English is not your native language. –  Sep 08 '18 at 13:05
  • Yes indeed but I don’t think the point is that here. The answer to your question is no of course since the result comes directly from at least two other results. It can be considered as a simple corollary. – user Sep 08 '18 at 13:08
  • @gimusi: You may as well be speaking Esperanto. –  Sep 08 '18 at 13:14

2 Answers2

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This is a particular case of Euclid's Proposition 38 from Book I of the Elements:

Triangles which are on equal bases and in the same parallels equal one another.

[ EDIT ]   Credit goes to @Micah's comment for pointing out that this particular case makes in fact the object of Proposition I.37:

Triangles which are on the same base and in the same parallels equal one another.

dxiv
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It is a simply application of the formula for the area of the triangle that is

$$S=\frac12\cdot AB \cdot d$$

where $AB$ is the length of the segment between the two fixed point and $d$ is the distance between the two lines.

That property can be viewd also as a particular case of Cavalieri's Principle

enter image description here

user
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  • How do you know that all three segments, at a given height in the above diagram, have the same length? –  Aug 16 '18 at 13:46
  • @EulerSpoiler it is explained in the given link, take a look here https://meangreenmath.com/2013/10/02/area-of-a-triangle-equal-cross-sections-part-2/ – user Aug 16 '18 at 14:14
  • Okay, thanks. I took a tl;dr stance first time around, but now I see where you're coming from. –  Aug 16 '18 at 14:38