# Tag Info

### 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' ...

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

Parseval identities in the theory of Fourier series and integrals.

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### 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 $A$...
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### 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. ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 +...
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### 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 ...
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