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"All triangles are isosceles" is a famous geometric fallacy (see below). Unlike many other fallacies its flaw is subtle and hard to spot, so it is often used as a cautionary example against the "danger in diagrams", e.g. in Greenberg's text, but always, it seems, without attribution. It feels so much in the spirit of Euclid that one would think it may go back to antiquity, and Euclid did write a book of fallacies called Pseudaria, now lost. But no, only four geometric fallacies survive from antiquity, and "all triangles are isosceles" is not one of them.

So where did it come from, who came up with it? Or if that can not be traced, what is the earliest known appearence? I am also interested in tracking the origins of other geometric fallacies.

"Proof":

All triangles are isoceles If the angle bisector at A and the perpendicular bisector of BC are parallel, then ABC is isosceles. On the other hand, if they are not parallel, they intersect at a point, which we call P, and we can draw the perpendiculars from P to AB at E, and to AC at F. Now, the two triangles labeled "alpha" in this figure have equal angles and share a common side, so they are equal by angle-side-angle. Therefore, PE = PF. Also, since D is the midpoint of BC, the triangles labeled "gamma" are equal right triangles by side-angle-side, and so PB = PC. From this it follows that the triangles labeled "beta" are right triangles with equal leg and hypotenuse, so equal to each other. Thus, we have BE+EA = CF+FA, meaning the triangle ABC is isosceles. (From mathpages)

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  • $\begingroup$ I'm guessing the fallacy is that P is actually outside the triangle? $\endgroup$ – IanF1 Jan 3 '15 at 12:17
  • $\begingroup$ The fallacy is actually deeper than that. If E fell on AB, and F on AC, the fallacy would still "follow", whether P were inside or outside triangle ABC. If E fell on AB produced, and F on AC produced, the fallacy would "follow", with the change that we subtract BE from AE and CF from AF. The truth is that, of E and F, one falls on a side, but the other falls on the production of a side. If, for example, E falls on AB produced, but F falls on AC, then AB=AE-BE but AC=AF+FC. $\endgroup$ – Rosie F Nov 28 '16 at 10:27
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This is in Rouse Ball's Mathematical recreations and problems (2nd edition, 1892, p. 33), and later editions carry this footnote (6th edition, 1914, p. 45):

I believe that this and the fourth of these fallacies were first published in this book.

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    $\begingroup$ You should state that the "fourth of these fallacies" is the isosceles fallacy discussed in this question. Nice answer! $\endgroup$ – Rory Daulton Jan 1 '15 at 18:20
  • $\begingroup$ Nice! I just had to look up this book for another question hsm.stackexchange.com/questions/751/… but missed this. I also found the fallacy described in Klein's Elementary Mathematics from an Advanced Standpoint: Geometry published in 1908, he just says "I'll gladly give an example that you are all probably familiar with". Authors of fallacies are forgotten quickly. $\endgroup$ – Conifold Jan 2 '15 at 20:55

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