New answers tagged geometry
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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
&...
5
There can be no standard algorithm to solve angle hunting problems. You have to build this skill in students by exposing them to simpler problems and gradually increasing the complexity. I suppose the difficulty of these problems can be guaged by the number of properties/constructions required to solve them completely.
The key thing is to begin by working ...
3
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.
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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 $|BC|$ ...
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, ...
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 $\frac{a^2+b^2+c^2}{6}+\frac{2\Delta}{\sqrt{3}}$;...
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 ...
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 $...
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.
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 ...
1
Are proofs still part of the geometry curriculum? (Some of my math colleagues have mentioned they've been downplayed over the last decade or so.) A good Chemistry answer looks an awful lot like a good proof; same sort of logic flow. (As does a good programming solution, if you're looking for another connection.)
It might be a bit simple, but you could ...
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 ...
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 ...
6
Pick your battles. Don't expect to have synergy in every place. But where you do have synergy, exploit that, call it a win, and move on. Concentrate on the partial fullness of the glass, not the partial emptyness. For that matter, you don't have time to totally redesign each course from the ground up in a way new to man. Nor do you want to screw up the ...
4
This is perhaps more molecular biology than it is chemistry.
There are some accessible planar geometric questions suggested by the
H-P (hydrophobic-hydrophilic) model of protein (amino acid) folding,
which could
be explored with simple manipulatives (such as K'nex).
For example, which proteins in this model have a unique minimum energy
folding?
&...
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One angle you could look at is molecular geometry. Not really my subject area but a couple of examples:
Organic molecules can have different chiralities. That means that while one is the mirror image of another you cant rotate one molecule to the other. The reasons for this are pretty deep mathematically, but chemically give rise to interesting things as ...
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