Questions tagged [proof-without-words]

For questions concerning the creation and understanding of pictorial proofs.

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Is there a geometric proof of $\frac1r = \frac{1}{h_a} + \frac{1}{h_b} + \frac{1}{h_c}$ in a given triangle?

Can anyone provide a geometric proof of $\frac1r = \frac{1}{h_a} + \frac{1}{h_b} + \frac{1}{h_c}$ in a given triangle, if there is one? Here, $h_i$ refers to the distance from the side $i$ to the ...
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2answers
59 views

Is there anything missing in this proof?

I came with this geometry problem and numerous lengthy solutions were proposed, so I thought there must be something missing in my solution. The problem: Given that $\angle CAB=3x$, $\angle BCA=x$, ...
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1answer
25 views

Is there a dissection proof of the Pythagorean Theorem, such that a single cut makes the needed pieces from the squares?

Can you cut the squares of a right triangle with one cut (cutting all three pieces at the same time) so that it’s possible to arrange them and prove Pythagoras’ theorem?
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1answer
118 views

Looking for an intuitive explanation of the gamma function which can be comprehended with high school maths. [closed]

I am looking for an "intuition pump" (Daniel Dennett's phrase) for the gamma function, showing why $\Gamma(n + 1)$ is the same as $\int_0^\infty x^n e^{-x} \, dx$. For instance, is the fact that $e^{...
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6answers
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Is there a sufficiently reachable plausibility argument that $\pi$ is irrational?

I was teaching someone earlier today (precisely, a twelve-year-old) and we came upon a problem on circles. Little did I know in what direction it would lead. I was able to give a quick plausibility ...
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0answers
30 views

Four Dimensional Geometric Representation of Sums of Cubes

Let $S=1+ 2^3 +\dots +n^3$. I have seen an assertion (Geometric interpretation for sum of fourth powers) that one can geometrically see the formula $S=\left[\frac{n(n+1)}{2}\right]^2$ by ...
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2answers
240 views

Quadrilateral in a Parallelogram - Interesting Proofs!

Here's an interesting problem, and result, that I wish to share with the math community here at Math SE. I think I've found a proof without words... I came across this problem sometime back, and here'...
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1answer
123 views

Geometric proof of a trig identity on $\cos t \cos u\cos v$

Consider the following trigonometric identity, valid for any set of angles $u,v,t$: $$\cos t⋅\cos u⋅\cos v =\frac14\left[\cos(t + u + v)+\cos(t + u - v)+\cos(u+v-t)+\cos(v + t - u)\right]$$ This ...
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3answers
100 views

Proof without words: $1+8\times\text{triangular number}$ is an odd perfect square

A recent question asked how to show that $8T_n+1$ is a perfect square if $T_n$ is a triangular number. This follows immediately from $T_n=\frac12 n(n+1)\implies 8T_n+1=4n^2+4n+1=(2n+1)^2$. Can this ...
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3answers
191 views

Proof of four triangles with equal area.

I have been working on proofs without words but I have a bit problem about Steven Snovers proof. It proves that these four triangles have equal area. Proof comes from rotation but I couldnt get it.
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1answer
168 views

Proof without words for Stirling formula

It it easy to prove $$x^n=\sum_{i=0}^{n}S(n,i)x_{(i)}$$ by induction or recursive formula... But is there some proof without word for it? or at least a direct proof that concern with partitions of ...
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4answers
5k views

What is the explanation for this visual proof of the sum of squares?

Supposedly the following proves the sum of the first-$n$-squares formula given the sum of the first $n$ numbers formula, but I don't understand it.
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4answers
116 views

To prove $\frac{n!}{p!q!}$. [closed]

The number of permutaion of n objects, where p and are of one kind,q are of second kind,are of different kind is $\frac{n!}{p!q!}$ How can we proove above theorem $\frac{n!}{p!q!}$.I tried to prove ...
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2answers
141 views

Examples of 3d visual proofs

I am looking for examples of three dimensional constructible proofs. By this I mean activities such as steps in proving $1^2+2^2+\cdots+n^2=n(n+1)(2n+1)/6$. In this construction the identity is proven ...
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4answers
242 views

Geometric proof of $QM \ge AM$

Prove by geometric reasoning that: $$\sqrt{\frac{a^2 + b^2}{2}} \ge \frac{a + b}{2}$$ The proof should be different than one well known from Wikipedia: DISCLAIMER: I think I devised such proof (...
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0answers
198 views

Using a visual “proof” to show that $\sum_{n=1}^{\infty} \left(\frac 34 \right)^n =1$

The "proof without words" that $\sum_{n=1}^{\infty} \left(\frac 12 \right)^n =1$ is fairly well known: But why can't we apply the exact same logic to $\sum_{n=1}^{\infty} \left(\frac 34 \right)^n$ or ...
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2answers
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Visual proof of $\sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}$?

In his gorgeous paper "How to compute $\sum \frac{1}{n^2}$ by solving triangles", Mikael Passare offers this idea for proving $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$: Proof of equality ...
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1answer
176 views

On equilateral triangles

I came across the following geometry problem. In the exterior of a triangle $ABC$ three equilateral triangles $ABC' , BCA'$ and $CAB'$ are constructed. Prove that the centroids of these triangles ...
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2answers
359 views

Explain this calculus proof without words

This demonstrates that $\int_0^1 t^{p/q} + t^{q/p} dt = 1$. Could you please explain how the proof without words shows that?
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1answer
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I can't figure how to prove this coordinate algebraic geometry proof.

Consider the points $A$ and $C$ are on $y=x^p$. We are told that point $A$ has coordinates $( a, b )$, where $0<a<1$ and point $C$ has coordinates $( c, d )$, where $1<c$. Give a ...
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0answers
310 views

Looking for proof-without-words of Bezout's identity

I'm looking for a "proof-without-words" of Bezout's identity (for integers). Does anyone know of one?
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1answer
227 views

Does $\operatorname{MSE}(\hat{\theta}) = \operatorname{Var}(\theta)+ \left(\operatorname{Bias}(\hat{\theta},\theta)\right)^2$?

We know that $\operatorname{MSE}(\hat{\theta})=\operatorname{E}\left[(\hat{\theta}-\theta)^2\right]$ and $\operatorname{MSE}(\hat{\theta})=\operatorname{Var}(\hat{\theta})+ \left(\operatorname{Bias}(\...
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0answers
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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 ...
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2answers
918 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 ...
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2answers
246 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?...
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2answers
11k views

Geometric interpretation for sum of fourth powers

Summing the first $n$ first powers of natural numbers: $$\sum_{k=1}^nk=\frac12n(n+1)$$ and there is a geometric proof involving two copies of a 2D representation of $(1+2+\cdots+n)$ that form a $n\...
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4answers
538 views

Proof-without-words for $\bar a\times (\bar b\times\bar c)=\bar b (\bar a\cdot\bar c)-\bar c (\bar a\cdot \bar b)$ or some visual-biased explanation?

Griffiths' Introduction to Electromagnetism -book has equations called 20.10 below. I have proved this equation d) pretty much on the first mathematics -course I had but I have not yet understood a ...