User:MRFS/sandbox

Duality

From the above formula the height H of the outer Napoleon triangle may be written H = $\scriptstyle {\sqrt {{\frac {1}{8}}(a^{2}+b^{2}+c^{2}+4\Delta \surd 3)}}$ where Δ is the area of the triangle with sides $a,b,c$. The inner triangle has height $\scriptstyle {\sqrt {{\frac {1}{8}}(a^{2}+b^{2}+c^{2}-4\Delta \surd 3)}}$ from which it follows that the difference in area of the two Napoleon triangles is equal to Δ.

Several seemingly unrelated questions share the same answer namely 2H, and this coincidence is sometimes referred to as duality.

Q1. Find the height of the largest equilateral triangle that circumscribes a given triangle with sides $a,b,c$.

Q2. Find the side of an equilateral triangle in which the distances from an internal point to its vertices are $a,b,c$.

Q3. Given a triangle with sides $a,b,c$ in which no angle exceeds 120°, find the minimal sum of the distances from any point to the vertices.

In fact 2H is the algebraic sum of the distances from Triangle center X13 to the vertices. So whenever X13 lies outside the triangle the distance from it to the nearest vertex must be taken as negative. (Nearest argument :- If A > 120 let XA cut BC at Y. Then BXY = 60 = YXC so B < 60 and AYB > 60 which means XA < XY < XB)

The Fermat point determines the answer to Q3. It was raised by Fermat in the early part of the 17th century as a challenge to his peers. A particular case of Q1 can be traced back to page xx of the 1755 Ladies' Diary. The notion of duality has been dated back to 1812.

NARRATIVE On reflection I now think that Q1 (probably) and Q2 (possibly) also require the largest angle restriction. After a lot more reflection I've decided the restriction isn't needed. In Q2 the fact that the point is internal forces a,b,c to form a triangle where no angle exceeds 120°. In Q1 note that if X is the first isogonic centre the formula gives the algebraic sum of AX+BX+CX and not |AX|+|BX|+|CX|. So if A is internal to XBC then AX must be treated as negative. This is exactly what is needed since it is the algebraic sum of AX+BX+CX that relates to H. The statement in Kuhn's article does need qualification. In the "flat" 2,1,1 triangle the minimum sum is 2 whereas the maximum height is root3.

Find a point that minimises the sum of the distances from it to three given points.

Q1 is answered by the Fermat point. The minimal sum is 2H so long as the three given points do not form a triangle with an angle exceeding 120°. 2H is also the answer to Q3 whilst the answer to Q2 is 2L.

The height H of the Napoleon triangle is $\textstyle {\frac {\surd 3}{2}}$ L .

1⁄2   L⁄2   ½L√3. L√3⁄2

This is most annoying apparently ttp://www.convexoptimization.com/wikimization/index.php/Fermat_point with an h at the front has been blacklisted by Wikipedia which means it can't be used. my attempted edit of related problems has been completely destroyed. let's try again but first save this edit.

Duality

well, that confirms the said page is blacklisted but the alternative page shown above isn't as pretty and not one i would wish to include. what a pity. where do we go from here?

it would seem that the wikimization site is regarded as a possible rival which may be why it is blacklisted. at present it doesn't look like a serious rival as it doesn't know about napoleon, morley, perron-frobenius, etc, etc. wikipedia itself doesn't mention it so possibly it's just another good idea that went nowhere. that being the case i clearly can't use it.

There are many vexed typesetting problems here. Basically there are only 2 options, wiki-markup and latex, but unfortunately neither of them is ideal.

Are we still in TNR? Yes! Subscripts are a_ij ....

Now let's adjust the size .....

There are many vexed typesetting problems here. Basically there are only 2 options, wiki-markup and latex, but unfortunately neither of them is ideal. Latex text is far too big for inline maths and needs reduced in size by \scriptstyle which puts it slightly below the rest. The correct vertical alignment requires the style="vertical-align:-xx%;" construct. And if this is ever fixed (and it is being worked on) then the fix could upset the alignment. Advice seems to be to use wiki-markup italics instead but that often produces poorish math displays. &...; is a potentially useful function as it can theoretically generate more or less any Unicode character, but there's a suggestion that those actually available in practice depend on the browser. A comparison with Google Docs is interesting. It too will generate Unicode as special characters but it doesn't have an equation editor. Nevertheless an equation editor (preferably LaTeX) is high on various wish lists so there must be a good chance that one will surface sooner or later. a_ij

A useful reference is http://en.wikipedia.org/wiki/Mathematical_HTML. This shows how to change font sizes ∑Psizes and Psizes and spacing (thinsp, ensp, emsp, nbsp).

i guess $x=-b\pm b^{2}-4ac$ is how we get inline matrices/formulae into wiki x = − b ± √b² − 4ac   ∑cos A

x = 0 x = -bpm b²−4ac what is going on - maybe the MATH here is what i want - at least it displays the right size - but note that the equals sign must be done in the way shown or it won't display at all.

Most LaTeX functions eg $\pm$ seem to trigger large fonts by default but others like $\sin$ don't.

Math versions are $\sin ^{2}2B$ - why not in big font? α β γ αβγ ½(a+b+c)

A further take on the vertical alignment is that it will render slightly differently under different browsers (Internet Explorer, Firefox, Chrome). This means there is absolutely no point trying to fix the formatting in this article until better software gives a consistent uplift. But check what I did with the Fermat Point article as I recall that is much improved (cut out all \math formatting).

However the general advice seems to be to use wiki-markup for in-line work and \math for display. Sadly wiki-markup is so bad for summations and subscripts (see below) that this just won't work.

Important note : It is possible to produce decent summation signs without using math. The trick is to force a common font such as Times New Roman that does display them correctly. It will take some time to assess if this could be worthwhile. Can whole articles be switched to TNR?

The general construct is Φ(A) Σ

An example is Σ ABC Σ DEF Σ GHI

Note how the above constructs work. The font style commands are cumulative and individual /font commands are needed to cancel them!

&#1D6BA; 𝚺 1D6BA ∑ Σ ∑

Which is best cos(A) + sec(B)sec(C) or cosA + secB secC or cos A + sec B sec C ? Strangely I think I prefer the middle one! Also need to consider sin2B , sin 2B , sin²2B , sin² 2B , sin ² 2B , sin ²2B

Which is best cos(A) + sec(B)sec(C) or cos A + sec B sec C or cos A + sec B sec C ? Strangely I think I prefer the middle one! Also need to consider sin 2B , sin 2B , sin² 2B , sin² 2B , sin ² 2B , sin ²2B

Math versions are $\sin ^{2}2B$ - why not in big font?

Before I can move forward I have to find where TT heard about connected matrices as I can't find any references to them. The term irreducible seems to be preferred but in my opinion is grossly overused. Then again M&M uses indecomposable instead yet connected is far more intuitive. But decomposable is more commonly taken to refer to Jordan/Spectral decomposition. What a mess! See M&M page 122 for its notation. Denote the (i,j) entry of A^k by $\scriptstyle a_{ij}^{(k)}$ as opposed to my $\scriptstyle [A^{k}]_{ij}$.

It turns out that the presentation of matrices is a complete mess. Here are a few attempts at it. Find a projection with at least one zero but less than a block of them. The purpose is to show that "irreducible" must be qualified by "non-negative" in order for a block of zeros to exist. The Wikipedia definition of irreducibility actually requires the assumption of non-negativity in order to work; that is assuming such a projection exists (which i'm pretty sure of). And here is one ...

$\left({\begin{smallmatrix}\quad 0&\;\;\;\,\,0&-33&-33\\-10&\;\;\,10&-41&\;\,-1\\\;\;\,14&-14&\;\;64&\;\;\;\,8\\-14&\;\;\,14&\;\;\;2&\;\,\,58\end{smallmatrix}}\right)$

For example the projection P = $\left({\begin{smallmatrix}3&-4&2&2\\3&-5&3&3\\3&-4&2&2\\0&-2&2&2\end{smallmatrix}}\right)$ has only the one zero and so is irreducible.

$\left({\begin{array}{rrrr}\scriptstyle 0&\scriptstyle 0&\scriptstyle -33&\scriptstyle -33\\\scriptstyle -10&\scriptstyle 10&\scriptstyle -41&\scriptstyle -1\\\scriptstyle 14&\scriptstyle -14&\scriptstyle 64&\scriptstyle 8\\\scriptstyle -14&\scriptstyle 14&\scriptstyle 2&\scriptstyle 58\end{array}}\right).\left({\begin{array}{rrrr}0&0&-33&-33\\-10&10&-41&-1\\14&-14&64&8\\-14&14&2&58\end{array}}\right).$

The font here is too large but I can't seem to alter it. Using "smallmatrix" instead of "array" works but then {rrrr} doesn't. \scriptstyle has no effect or causes an error. This is all so silly. asserts that a real square matrix with strictly positive entries has a unique largest real eigenvalue and that the corresponding eigenvector has strictly positive components.

$\scriptstyle \lceil$ \\ $\scriptstyle \vert$ \\ $\scriptstyle \vert$ \\ $\scriptstyle \lfloor$	0	0	-33	-33	${\begin{cases}\\\\\\\end{cases}}$
	-10	10	-41	-1
	14	-14	64	8
	-14	14	2	58

$\left ($	0	0	-33	-33	$\right )$
	-10	10	-41	-1
	14	-14	64	8
	-14	14	2	58

	${\begin{cases}\\\\\end{cases}}$	1	if a2 > b2 + bc + c2	(equivalently A > 2π/3)
Let     f(a,b,c)     =		0	if b2 > c2 + ca + a2 or c2 > a2 + ab + b2	(equivalently B > 2π/3 or C > 2π/3)
		csc(A + π/3)	otherwise	(equivalently no vertex angle exceeds 2π/3).

real numbers are $\mathbb {R} ^{\mathrm {3} }$ or R³.

real numbers are R³ or ℝ³.

real numbers are R³.

real numbers are R³.

normal size real numbers are ℝ³.

Greek and Coptic

i guess           $x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}$           is how we get inline matrices/formulae into wiki

but what about google? a ≥ b