Mathematical trap: All triangles are equilateral

Challenge: look at this video and evaluate its reasoning, considering its strengths, weaknesses and flaws. Can you evince that all triangles are equilateral?

Try it before reading the next section.

Enigma excavated

Is the reasoning any compelling? For outsiders, they would think it as alchemy, as the reasoning seems plausible: oh yeah the length from any point on the bisector line must be equal to the two sides, and the triangles XXX must be similar, and…(after a series of reasoning) the three sides must be equal. But any insider (including amateur ones like me) would excoriate it, even at the middle of the video (2 minutes+). Why?

The blunder was subtle, being well-muffled by the apparent logic. But it can be egregious in this context. This tiny little flaw fumbled the whole argument, and defiled the beauty of geometry: yes the distances from point to two sides are equal and the length from the same point to the vertices are also equal, but the perpendicular feet are of different nature: one on the side and the other on the extension! Try to factor in my argument, and it would be overt that the statement contravenes the mathematical principles.

In general, all problem solving (not only mathematics, but also other arguments in real life) are prone to these pitfalls, here and there. For mathematical part, Let me share with you this note on geometry, obtained on Canadian training website (Winter 2011):

The notes draw interests by beginning with admonition on pitfalls before readers commit the same mistake. A typical (and one of the most famous) example was IMO 2009 Problem 4, where the shenanigan in the problem ruined contestants who were delinquent about the case finding. The other case was IMO 2012 Problem 4 (again!)–functional equations. The enormity deception of the problem, along with the atrocity of marking scheme, eradicated a great crowd of the contestants including 2 from Malaysia. While I was not affected, the problem ebbed my potential on solving problem 5, simply because I used 3.5 hours on that single problem (even longer than problem 2). Along with checking process, that left me 15 minutes before the bell rang. The performance of teams was, unfortunately, eclipsed by it, especially when it was just problem 4.

So in the end, what is the moral of the story? Make sure your cases are exhaustive, or you will be entrapped. How about in real life? It’s equally prevalent, considering the overwhelming number of con mans in the market. It’s such a bliss that I take up Thinking Skills lesson for AS level, but it was unfortunately that, my ability in reasoning is confined to mathematics, and had hard time in dealing with Thinking Skills questions (esp paper 2). Note: the challenge statement in the first sentence in this post is actually a Thinking Skills question, usually in paper 2, 3(b). Nevertheless, a good skill to pick up.

Disclaimer: The 2nd paragraph of the 2nd section doesn’t have any intention to chastise the persona in the video presenter. Instead, his efforts in entertainment should be appreciated. However, if any student comes out of this fallacious argument, then he deserves to be punished by coordinators.

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