34
votes

### Which theorems have Pythagoras' Theorem as a special case?

The Law of cosines is the first that comes to my mind:
$$c^{2}=a^{2}+b^{2}-2ab\cos \gamma$$
(source: Wikipedia)
If $\gamma$ is a right angle, its cosine is 0 and all that remains is Pythagoras' ...

Community wiki

29
votes

### Which theorems have Pythagoras' Theorem as a special case?

Parseval identities in the theory of Fourier series and integrals.

Community wiki

22
votes

### Which theorems have Pythagoras' Theorem as a special case?

The parallelogram law says that if $\mathcal{V}$ is an inner product space and $\mathbf{v},\mathbf{w} \in \mathcal{V}$, then
$$
2\|\mathbf{v}\|^2 + 2\|\mathbf{w}\|^2 = \|\mathbf{v} + \mathbf{w}\|^2 + \...

Community wiki

22
votes

### Which theorems have Pythagoras' Theorem as a special case?

The Pythagorean theorem is a limit of the general formula for spherical or hyperbolic space: $\cos(a\sqrt{\kappa})\cos(b\sqrt{\kappa})=\cos(c\sqrt{\kappa})$, where $\kappa$ is the curvature.

Community wiki

21
votes

Accepted

### Is there a pyramid with all four faces being right triangles?

This picture of a quadrirectangular tetrahedron is from István Lénárt, `The Right Triangle as the Simplex in 2D Euclidean Space, Generalized to $n$ Dimensions'

18
votes

Accepted

### The 4th vertex of a triangle?

I propose an alternative "fourth vertex".
To paraphrase Sherman's result:
The "fourth side" ($w$) of a triangle is a chord of the circumcircle, is a tangent to the incircle ($\...

17
votes

### Packing an upwards equilateral triangle efficiently by downwards equilateral triangles

OK, posting then.
I prefer to think of triangles pointing to the right in the triangle pointing to the left. Let $\delta=e^{-\sqrt{\log 1/\varepsilon}}$. For each small triangle $T$, let $I$ be the ...

16
votes

Accepted

### Three circles intersecting at one point

Such things are quick in complex numbers. Let $O=0$ be the origin, $ABC$ be the unit circle. The centroid of $ABC$ is $G=(A+B+C)/3$, the Euler circle is the image of the circle $ABC$ under homothety $...

16
votes

### Which theorems have Pythagoras' Theorem as a special case?

This is probably the simplest: $(a+b)^2=a^2+b^2+2ab$, if you take $a,b$ elements of some inner product vector space and $(\cdot)^2$ means inner product with itself.

Community wiki

13
votes

### Two queries on triangles, the sides of which have rational lengths

On the second question: There is the inequality $12\sqrt{3}A\leq P^2$ that needs to be satisfied first of all to get a triangle.
Using Heron's formula for a triangle with sides $x$ and $y$, you are ...

13
votes

### Which theorems have Pythagoras' Theorem as a special case?

Pythagoras' theorem is a special case of the three point identity for Bregman distances: Let $h$ be convex and lower semi-continuous on a Banach space - further assume differentiability of $h$ for ...

Community wiki

12
votes

Accepted

### Do two new special points in any triangle exist?

Your conjecture is true: these two points are sometimes called the equicevian points of $ABC.$

12
votes

### Distance between point inside a triangle and its vertices

Let $a, b, c$ be the side lengths of the triangle, and $x, y, z$ the distances from a point inside a triangle to the respective vertices. Then the numbers $x,y,z$ satisfy the equation
$$\begin{vmatrix}...

12
votes

Accepted

### Intersection point of three circles

The points $A,B,C$ are midpoints of the sides of $\triangle A'B'C'$, thus $H$ is the centre of the circumcircle $\omega$ of $\triangle A'B'C'$. Make an inversion with respect to $\omega$. The point $A$...

11
votes

### Which theorems have Pythagoras' Theorem as a special case?

One of the most attractive generalisations is de Gua's theorem: If a tetrahedron $ABCD$ is rectangular at $A$, then $$ |BCD|^2=|ABC|^2+|ACD|^2+|ABD|^2$$ where the absolute value signs denote area.
...

Community wiki

11
votes

Accepted

### Packing an upwards equilateral triangle efficiently by downwards equilateral triangles

Ok I think this is an argument that the perimeter at least $\varepsilon^{-c}$ for some sufficiently small $c$. To avoid special cases where $\varepsilon$ is large, we use the convention that we count ...

10
votes

Accepted

### On 4 random points in a rectangle

As explained in Square Triangle Picking, the mean area of a triangle picked inside a rectangle of unit area is 11/144. So the probability that the fourth point lands inside this triangle is $11/144=0....

10
votes

### Which theorems have Pythagoras' Theorem as a special case?

$$1=\cos^2 x+\sin^2x$$ (which can be proven without using Pythagoras) holds for arbitrary $x$ in $\mathbb C$ and yields Pythagoras for real $x$.

Community wiki

10
votes

### Which theorems have Pythagoras' Theorem as a special case?

I like to think about Pythagoras theorem as a corollary/special case of the following theorem:
Theorem: Let $X$ be a finite dimensional real Banach space such that the group of linear isometries (that ...

Community wiki

9
votes

### Six points on an ellipse

It is easy to see that $IJ$ is parallel to $AB$, etc. The result follows from the converse of the Pascal's theorem: Consider the hexagon $MHKLIJ$, then the intersection points of the three pairs of ...

9
votes

### Need help with finding all angles of 11 sided 3D object

There is a problem in the design,
if I understand the phrases "the height I want the spire to be"
and "how tall I choose to make the spire":
these suggest there is freedom to ...

9
votes

### Which theorems have Pythagoras' Theorem as a special case?

The Binet-Cauchy formula says that if $A$ and $B$ are a $n\times m$ and $B$ a $m\times n$ real matrices, respectively, and for $s\subseteq\{1,\ldots ,m\}$ with $|s|=n$ we denote by $A_s$ the $n\times ...

Community wiki

8
votes

### Do two new special points in any triangle exist?

These two points are the focuses of Steiner circumconic https://en.m.wikipedia.org/wiki/Steiner_ellipse.

8
votes

### Why are the medians of a triangle concurrent? In absolute geometry

Below is a summary of Hjelmslev's argument as it is explained in Bachmann's book, see the references to German and Russian editions in Misha's answer. The goal of this answer is at first to help ...

8
votes

Accepted

### Three circles meet at a point

Let $P$ be the image of $O$ by the inversion with respect to the incircle. Since $IA\cdot IA'=IO\cdot IP=r^2$, we have that $A, A', O, P$ are concyclic, so $P$ is on $k_1$. Similarly $P$ is on the two ...

8
votes

### Which theorems have Pythagoras' Theorem as a special case?

The discrete form of the parallel axes theorem for the second moment of area for $\,n\,$ points $\,A_k\,$ with centroid $\,G\,$ and an arbitrary point $\,P\,$ is $\,\sum_{k=1}^n PA_k^2 = n \cdot PG^2 +...

7
votes

Accepted

### Why are the medians of a triangle concurrent? In absolute geometry

There is such a proof given by Hjelmslev; it is based on a clever application of central symmetries (point-reflections). You can find it on pages 102-104 of
Ф. Бахман, Построение геометрии на основе ...

7
votes

### Maximizing the area of a region involving triangles

I find a larger area than @MattF in a non-symmetric solution. A numerical approximation to that solution is:
...

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