27

Given that I had a good time coercing Mathematica to give me the pictures below I might as well promote my comment to an answer. My favorite such a result is the so called Steiner's porism. It is a statement about the relation of two circles, one inside the other. Such as these two: We get a so called Steiner chain of circles from this by first drawing a ...


18

I'm not sure whether your question aims at educators or at the students themselves, but my answer actually would be very similar. Having as well as being a good, perhaps gifted student is a treasure. It's fun both to teach and to be one. The subject is fun. Conversations are fun. New angles are appearing which were not obvious, perhaps not even to the ...


13

I'm going to rewrite this answer to clarify what I think the issue is. I think the OP is imagining a different definition of the ring $k[x]$ than most answerers are. Here are two reasonable definitions: $k[x]$ is the ring of formal expressions of the form $\sum_{j=0}^{\infty} p_j x^j$ with $p_j \in k$ and we require that $p_j$ is $0$ for $j$ sufficiently ...


12

Although I was shown the theorems of Ceva and Menelaus in high school, I don't think they're part of the standard curriculum. And I consider them beautiful, especially when considered together so that their similarities become apparent. I'd also be inclined to nominate Desargues's theorem, especially in view of the observation that the Desargues ...


11

Short answer: The skewed content is not a good reason for avoiding IMO-style contest training, because if the training is done right then the students will be led to explore mathematics and would never have a mistaken picture that mathematics is mostly about IMO topics. (I of course compare between good IMO-style training and good teaching of university-...


10

One rather major argument in favour of contest-style mathematics is its ability to cultivate problem-solving abilities in students while not requiring much difficult machinery. It is of course undeniable that the ways of thinking in contest math are much different from the ways of thinking in university mathematics, but what both have in common is a ...


9

Not quite as beautiful as Morley’s trisection theorem, but here are two I never saw in high school but which I find beautiful: Ptolemy’s theorem, which says that for a cyclic quadrilateral with vertices on a circle, the sum of the products of opposite sides is equal to the product of the diagonals. A theorem of Pappus, which shows how to construct the ...


8

For the "beautiful" aspect, you might be inspired by other geometry theorems as illustrated by the 1960's painter Crockett Johnson. For example, here is the Morley Triangle, which is the first place I ever saw that particular theorem. Here is his painting of Pascal's Hexagon: When the opposite sides of a irregular hexagon inscribed in a circle are extended, ...


6

Pappus's Hexagon Theorem has a particularly nice diagram associated with it. Here, $A$, $B$ and $C$ can be any three points along the line $g$, while $a$, $b$ and $c$ can be any three points along the line $h$. The theorem states that the points $X$, $Y$ and $Z$ at the intersections all lie along a common line. This theorem was known in antiquity. What is ...


5

This does not directly address your question (and so is not a direct answer), in that I am not sure about the availability of solutions (Jiri, is ... no longer with us). But it is an impressive textbook, unusually concise (~200 pages). Matousek, Jiri, and Bernd Gärtner. Understanding and using linear programming. Springer Science & Business Media, ...


5

One thing that hasn't been mentioned yet is the social aspect. There are always some people who like to stay alone, but for the large majority, meeting like-minded people and engaging in some sort of activity with them is much more fun than sitting at home and staring at textbooks. Especially at the higher level, where some traveling is involved, ...


4

1. Fundamental theorem of transformational plane geometry A transformation $\psi$ is an isometry iff $\psi$ can be expressed as a composition of three or fewer reflections. 2. Laisant's theorem Given a triangle $ABC$, extend the two sides $AB$ and $AC$ as necessary to point $A_c$ on $AB$ at distance $|BC|$ from $B$ and $C_b$ on $AC$ at distance $|BC|$ ...


4

Van Aubel's theorem comes to mind for me: On each side of a planar quadrilateral (can be non-convex), construct a square (all external or all internal to the quadrilateral). If we construct line segments between the centers of opposite squares, then the two line segments are orthogonal to each other and have equal length.


4

I'd recommend literally anything in either of these books, but I'm also partial to Napoleon's theorem. In particular, we can prove: The equilateral triangles erected outward on the sides $a,\,b,\,c$ of an area-$\Delta$ triangle have orthocetres at the vertices of an equilateral triangle of squared side length $\frac{a^2+b^2+c^2}{6}+\frac{2\Delta}{\sqrt{3}}$;...


3

I don't know the answer, but clearly understand the question...which means it is a great question. I think it will depend both on the student (interests, abilities) and the situation. For the situation, it probably includes quality, but also pedagogy (efficient approach) as well as fun factor. Consider the difference between just having a library card ...


2

A simple Google search (linear programming problems and solutions textbook) yields this result as the first hit: https://www.springer.com/gp/book/9783540417446 (Linear Optimization and Extensions: Problems and Solutions; Authors: Alevras, Dimitris, Padberg, Manfred W.) This appears to be exactly what you are looking for, based on reading the intro as well ...


2

Take a circle of radius 1. Inscribe a regular triangle in this circle. Inscribe a circle in this triangle. Inscribe a square in it. Inscribe a circle, regular pentagon, circle, regular hexagon and so forth. The radius of the limiting circle is called the Kepler-Beoucamp constant.


2

Either 1 or 3, for the reasons given in the other answers. But I would also note that "as $x$ approaches $a$" need not be set aside by commas (or a break in speech), as it is not optional in the sentence. Besides the arrangement of the words, it is worth noting that ISO 80000-2, Mathematical signs and symbols to be used in the natural sciences and ...


2

I do not believe that this is a concern that would surface, or be worth surfacing, in the courses named by this question's title. In my reading, the question is analogous to worrying about whether you can ask about e.g. the thousands place of $7521$: To do so assumes that the base ten representation of $7521$ is unique, or, in particular, that "thousands ...


2

A triangle's altitudes concur. It seems that this theorem wasn't known until Chapple. Related to this: If $H$ is $ABC$'s orthocentre, $A, B, C, H$ are an orthocentric system, i.e. the straight line between any two of them is perpendicular to that between the other two; and each is the orthocentre of the triangle formed by the other three. Where $R$ is $...


1

I believe an appropriate strategy that will help students achieve the goals you state is inquiry based learning (IBL) (in math, and ideally, other subjects, as well). Outcomes were evaluated in this study/research on IBL in mathematics at the college level, which provides background describing IBL in mathematics, as well as outcomes from a two-year ...


1

I'm posting this for Joseph Malkevitch, who had difficulty while posting. He wanted to include one of his favorite theorems, the Wallace-Bolyai-Gerwien theorem, that says that any two polygons of the same area have a common dissection. See this impressive app developed by Satyan Devadoss's students: Scissors Congruence app         &...


1

Soon after introducing polynomials, students learn to add and subtract polynomials. Notice that if $f(x)$ and $g(x)$ are polynomials with degrees $n$ and $m$ respectively, and if $f(x)-g(x)=0$, then $n=m$. It follows that the degree of a polynomial is well-defined. So the proof that degree is well-defined is not difficult at all. On the other hand, students ...


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