# Questions tagged [proof-without-words]

For questions concerning the creation and understanding of pictorial proofs.

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### Is it possible to explain geometrically why $\arctan (1/2) +\arctan (1/3) = 45$ degrees?

$\arctan(1/2)$ seems to be some strange, irrational angle, and the same goes for $\arctan(1/3)$, but those two angles seem to sum up to $45$ degrees. This seems like a mystery to me even though I can ...
108 views

### Geometric proof that $1 + 2+ 3 + \dots + n=\frac{(n+1)n}{2}$

I understand that the sum $1 + 2+ 3 + \dots + n$ takes the form $\frac{(n+1)n}{2}$. This can be shown symbolically: $1+2+3+\dots+n$, written backwards is $n+(n-1)+(n-2)+(n-3)+\dots+3+2+1$. If you ...
73 views

### Showing in general way that three square pyramids can combine to form a cuboid having same dimensions as square pyramid Visually

I know that volume of a square base pyramid of bottom square area $S$ and height $H$ is given by $\frac{1}{3}SH$ , and volume of a square base cuboid outside having dimensions exactly that to be $SH$ ...
184 views

1 vote
271 views

### Regular hexagon divided into triangles

Problem: Give an regular hexagon and an interior point of this, join this point with each vertex. The hexagon is divided in $6$ triangles, paint the triangles alternately. Show that the sum of the ...
106 views

2k views

### Why matrix representation of convolution cannot explain the convolution theorem?

A record saying that Convolution Theorem is trivial since it is identical to the statement that convolution, as Toeplitz operator, has fourier eigenbasis and, therefore, is diagonal in it, has ...
1k views

### Intuitive/Visual proof that $(1+2+\cdots+n)^2=1^3+2^3+\cdots+n^3$ [duplicate]

$$(1+2+\cdots+n)^2=1^3+2^3+\cdots+n^3$$ I noticed this only because $\displaystyle \sum_{i=1}^n i = \frac{n(n+1)}{2}$ and $\displaystyle \sum_{i=1}^n i^3 = \frac{n^2(n+1)^2}{4}$. But the two things ...
1 vote
496 views

### If $Ax=b$, for $b\ne 0$, has more than one solution, then $Ax=0$ does as well. T or F. Prove this.

I get that this is true, because there's one free variable, so no matter what the augmented matrix is, there always will be an infinite amount of solutions. Right? But how to I explain this as a proof?...