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Recently I ran into professor Halfbrain on the street. The professor told me that he had been spending his time with cutting triangles into smaller triangles. The professor has made the following amazing discovery:

Professor Halfbrain's theorem:
Every triangle can be cut into exactly $77$ smaller triangles that all are isosceles.

Is the professor's theorem indeed true, or has the professor once again made one of his phenomenal mathematical blunders?


(Recall that an isosceles triangle is a triangle, in which two or all three sides have equal length. This puzzle only considers so-called non-degenerate triangles, that is, triangles in which all three sides have strictly positive length.)

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up vote 14 down vote accepted

The theorem is

True

In fact

Every triangle can be cut into exactly $n$ smaller triangles that all are isosceles where $n \ge 4$

Expalanation:

Every triangle can be cut into two right triangles by cutting it along the altitude to its longest side for example. And every right triangle can be cut into two isosceles triangles by cutting along the median of its right angle. Therefore, every triangle can be cut into 4 isosceles triangles.

Now, given any triangle you can always make a cut from a vertex so you end up with at least one isosceles triangle. The other triangle can be cut into 4 isosceles triangles as shown before, therefore, every triangle can be cut into 5 isosceles triangles. Using induction you can see that this way, you can get any $n$ triangles.

EDIT:

The equilateral is a corner case since you can't cut a isosceles triangle from it by cutting from a vertex. To get 4 triangles you can of course cut it into 4 other same sized equilateral triangles. To get 5 triangles you can first cut off a smaller equilateral triangle leaving a isosceles trapezoid. And because isosceles trapezoids have a circumscribed circle you can cut the trapezoid into 4 isosceles triangles from the center of that circle.

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1  
No time to work on this so I may be saying something stupid but I don't see how you can cut from a vertex of an equilateral triangle to make an isosceles triangle. – Hugh Meyers Mar 21 at 10:32
    
@HughMeyers, true. the equilateral triangle is a corner case, I will expand the answer to cover that also in a minute – Ivo Beckers Mar 21 at 10:33
1  
The centre of the circumscribed circle isn't necessarily inside the triangle. – JiK Mar 21 at 12:07
    
@JiK you're right. good point. I guess it's $n \ge 4$ then like I said earlier – Ivo Beckers Mar 21 at 12:13
    
I'm not seeing how this would work for $n=6$, or how you'd cut the trapezoid into $4$ isosceles triangles. – user2357112 Mar 21 at 19:25

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