Askey–Gasper inequality: Difference between revisions

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{{Merge to|Pons asinorum|discuss=Talk:Pons asinorum#Merger proposal|date=June 2013}}
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Afterwards, you'll be compensated for your time and money you possess put in.<br><br>Whilst you have to pay much more upfront, your give back in the purchase is going to be very much better. With maintenance, you could possibly go annually without having issues, and get that season accompanied by mostly pricey repairs. There are so many things that you should know just before you can actually  [https://www.youtube.com/user/deangraziosi/ Dean Graziosi] be truly successful in marketing property.<br><br>Every additional payment you are making will shorten the lifespan of the personal loan a little bit. You may pay out your spouse back again with profits from the transaction of the home. Crucial and excellent locations, marketing, and transformation of prospect clients are all aspects that you might want to perfect - and they tips can really provide you with a wonderful edge in learning more about the main factors of property offering.<br><br>If you can afford it, consider creating another repayment on your home loan each and every month. Finding the right industrial house for the organization must include determining your space requirements. Every added settlement you are making will reduce the lifespan of your own bank loan just a little. Very clear off of your counters. For first time traders and also experienced investors, it is advisable to come with an expenditure companion.<br><br>In the event that your bank card stability is creeping up and also you have trouble maintaining the payments, it's a great idea to stop producing fees. This is when you are taking out a home financing to purchase eighty percent of the home and an further financial loan to cover the other 20 percent. For those who have floor tile counters, thoroughly clean the grouting, or replace it entirely if it's tarnished or discoloured.<br><br>It may need time to locate a profitable possibility, and after buying a home, it could need improvements or redecorating. Real-estate offers will appear and disappear and if you compromise your feelings on virtually any package, you are going to certainly make imprudent alternatives or take more time on feel dissapointed about than company. You should always look for a home that could support the volume of room your organization utilizes generally or needs for suitable work and procedure.<br><br>The excess settlement will utilize straight to the primary of your own loan. Feng Shui might be a effective resource to help sell your home quickly. When hiring out a home, constantly involve maintenance expenses when trying to figure out your finances.<br><br>Make an effort to set-aside a month-to-month allowance for fixes even in case they have not transpired however. The additional repayment will use straight to the main of your respective loan. Remove from your kitchen and bathroom counters. Shell out unique focus on the front door of your property.<br><br>This is important as your income rate increases together with the more models you might be leasing out. Don't make use of your greeting card until your month-to-month equilibrium has become paid off. This could be sure that your relocate is successful and successful.<br><br>Keep your concentrate on the result, not the deal that you skipped. Getting property can be a very pricey process. Also, be sure that the surfaces are glowing. An 80/20 loan may be the solution to your problem if you cannot appear to be to create the money for an advance payment on the residence. Whenever you can afford to pay for it, try out producing an added settlement on your house loan on a monthly basis.<br><br>Find any locations of your house that are darkish and uninviting. Before you decide to placed your home out there to market, be sure to substitute your linoleum surfaces. One important suggestion to not forget when investing in commercial real-estate is to find a home with as numerous models as you can actually afford.<br><br>Include lighting by means of an attractive lamp, or boost pre-existing lighting by using a mirror. Reduce your costs in any respect that you can, and employ an alternate transaction strategy, so that you don't find yourself hitting or overextending the reduce on your own bank card. Buying real estate can be quite a tough proposal should you get all your other worries involved.<br><br>A partner can assist you counteract the cost of acquiring home by utilizing funds or credit rating. You will probably have to put plenty of energy to your new expenditure at the start.
In [[Euclidean geometry]], the '''isosceles triangle theorem''', also known as the '''''pons asinorum''''', states that the angles opposite the two equal sides of an [[isosceles triangle]] are equal. It is, in essence, the content of proposition 5 in Book I of [[Euclid's Elements]].
 
==''Pons asinorum''==
{{main|Pons asinorum}}
 
In medieval times, when [[geometry]] was one of the four subjects of the [[quadrivium]], this theorem acquired a reputation as an early yet difficult proposition and those who did not master its proof would be unable to go on to more advanced material. Perhaps because of this, the theorem became known as the ''pons asinorum'' or asses' bridge and is used as a metaphor for any early, difficult test.
 
== Proofs ==
 
{| style="float:right"
| [[Image:IsoscelesTriangleProofByProclus.svg|right|thumb|195px|Proclus' proof]]
|}
{| style="float:right"
| [[File:Euclid 1 5 en.svg|right|thumb|195px|Euclid's Elements Book 1 proposition 5; the [[pons asinorum]] ]]
|}
===Euclid and Proclus===
Euclid's statement of the theorem includes a second conclusion that if the equal sides of the triangle are extended below the base, then the angles between the extensions and the base are also equal. Euclid's proof involves drawing auxiliary lines to these extensions. But, as Euclid's commentator [[Proclus]] points out, Euclid never uses the second conclusion and his proof can be simplified somewhat by drawing the auxiliary lines to the sides of the triangle instead, the rest of the proof proceeding in more or less the same way. There has been much speculation and debate as to why, given that it makes the proof more complicated, Euclid added the second conclusion to the theorem. One plausible explanation, given by Proclus, is that the second conclusion can be used in possible objections to the proofs of later propositions where Euclid does not cover every case.<ref>Heath pp. 251–255</ref> The proof relies heavily on what is today called [[side angle side|side-angle-side]], the previous proposition in the ''Elements''.
 
Proclus' variation of Euclid's proof proceeds as follows:<ref>Following Proclus p. 53</ref>
Let ABC be an isosceles triangle with AB and AC being the equal sides. Pick an arbitrary point D on side AB and construct E on AC so that AD=AE. Draw the lines BE, DC and DE. Consider the triangles BAE and CAD; BA=CA, AE=AD, and angle A is equal to itself, so by side-angle-side, the triangles are congruent and corresponding sides and angles are equal. Therefore angle ABE = angle ACD, angle ADC = angle AEB, and BE=CD. Since AB=AC and AD=AE, BD=CE by subtraction of equal parts. Now consider the triangles DBE and ECD; BD=CE, BE=CD, and angle DBE = angle ECD have just been shown, so applying side-angle-side again, the triangles are congruent. Therefore angle BDE = angle CED and angle BED = angle CDE. Since angle BDE = angle CED and angle CDE = angle BED, angle BDC = angle CEB by subtraction of equal parts. Consider a third pair of triangles, BDC and CEB; DB=EC, DC=EB, and angle BDC = angle CEB, so applying side-angle-side a third time, the triangles are congruent. In particular, angle CBD = BCE, which was to be proved.
{{clear}}
 
===Pappus===
Proclus gives a much shorter proof attributed to [[Pappus of Alexandria]]. This is not only simpler but it requires no additional construction at all. The method of proof is to apply side-angle-side to the triangle and its mirror image. More modern authors, in imitation of the method of proof given for the previous proposition have described this as picking up the triangle, turning it over and laying it down upon itself.<ref>For example F. Cuthbertson ''Primer of geometry'' (1876 Oxford) p. 7</ref>
This method is lampooned by [[Charles Lutwidge Dodgson]] in ''[[Euclid and his Modern Rivals]]'', calling it an "[[Irish bull]]" because it apparently requires the triangle to be in two places at once.<ref>Charles Lutwidge Dodgson, ''Euclid and his Modern Rivals'' Act I Scene II §6</ref>
 
The proof is as follows:<ref>Following Proclus p. 54</ref>
Let ABC be an isosceles triangle with AB and AC being the equal sides. Consider the triangles ABC and ACB, where ACB is considered a second triangle with vertices A, C and B corresponding respectively to A, B and C in the original triangle. Angle A is equal to itself, AB=AC and AC=AB, so by side-angle-side, triangles ABC and ACB are congruent. In particular angle B = angle C.<ref>Heath p. 254 for section</ref>
 
===Others===
[[Image:IsoscelesTriangleProofTextbook.svg|right|thumb|195px|A textbook proof]]
A standard textbook method is to construct the bisector of the angle at A.<ref>For example J.M. Wilson ''Elementary geometry'' (1878 Oxford) p. 20</ref>
This is simpler than Euclid's proof, but Euclid does not present the construction of an angle bisector until proposition 9. So the order of presentation of the Euclid's propositions would have to be changed to avoid the possibility of circular reasoning.  
 
The proof proceeds as follows:<ref>Following Wilson</ref> As before, let the triangle be ABC with AB&nbsp;=&nbsp;AC. Construct the angle bisector of angle BAC and extend it to meet BC at X. AB&nbsp;=&nbsp;AC and AX is equal to itself. Furthermore angle BAX = angle CAX, so, applying side-angle-side, BAX and CAX are congruent. It follows that the angles at B and C are equal.
 
[[Adrien-Marie Legendre|Legendre]] uses a similar construction in ''Éléments de géométrie'', but taking X to be the midpoint of BC.<ref>A. M. Legendre ''Éléments de géométrie'' (1876 Libr. de Firmin-Didot et Cie) p. 14</ref> The proof is similar but [[side side side|side-side-side]] must be used instead of side-angle-side, and side-side-side is not given by Euclid until later in the ''Elements''.
 
== In inner product spaces ==
 
The isosceles triangle theorem holds in [[inner product spaces]] over the [[real number|real]] or [[complex number]]s.  In such spaces, it takes a form that says of vectors ''x'', ''y'', and ''z'' that if<ref>J. R. Retherford, ''Hilbert Space'', [[Cambridge University Press]], 1993, page 27.</ref>
 
: <math> x + y + z = 0\text{ and }\|x\| = \|y\|,\, </math>
 
then
 
: <math> \|x - z\| = \|y - z\|.\, </math>
 
Since
 
: <math> \|x - z\|^2 = \|x\|^2 - 2x\cdot z + \|z\|^2,\, </math>
 
and
 
: <math> x\cdot z = \|x\|\|z\|\cos\theta\, </math>
 
where ''θ'' is the angle between the two vectors, the conclusion of this inner product space form of the theorem is equivalent to the statement about equality of angles.
 
== Notes and references ==
{{reflist}}
 
*Euclid, commentary and trans. by [[T. L. Heath]] ''Elements'' Vol. 1 (1908 Cambridge) [http://books.google.com/books?id=UhgPAAAAIAAJ Google Books]
 
*Euclid, commentary by Proclus, ed. and trans. by [[Thomas Taylor (neoplatonist)|T. Taylor]] ''Elements'' Vol. 2 (1789) [http://books.google.com/books?id=2SBeEnT45gsC Google Books]
 
* '''Geometry Textbook – Standard IX''', ''[[Maharashtra]] State Board of Secondary and Higher Secondary Education'', Pune – 411 005, '''[[India]]'''.
* John Stillwell, ''The Four Pillars of Geometry'', [[Springer-Verlag|Springer]], 2005, page 24.
 
[[Category:Articles containing proofs]]
[[Category:Euclidean geometry]]
[[Category:Theorems in plane geometry]]

Latest revision as of 21:11, 10 January 2015

A would-be buyer will need to have a good idea in the accessible counter-top room, and also this is extremely hard when it's full of clutter. Even so, don't give up even though this can devote some time. That means it is possible to pay off the loan speedier, protecting probably lots of money in fascination repayments. Make sure it is properly lit, and this your home number is clearly noticeable.

These floors are extremely out-of-date and other possibilities, including porcelain tile and hardwood floors, give your own home a neat and nicely-managed physical appearance. Your own home will market speedier if visitors possess a ideal impression from the minute they go walking through the front door.

It means you can pay off the loan faster, preserving most likely lots of money in interest monthly payments. Difficulties will almost certainly arise at some time so you need to be well prepared. Afterwards, you'll be compensated for your time and money you possess put in.

Whilst you have to pay much more upfront, your give back in the purchase is going to be very much better. With maintenance, you could possibly go annually without having issues, and get that season accompanied by mostly pricey repairs. There are so many things that you should know just before you can actually Dean Graziosi be truly successful in marketing property.

Every additional payment you are making will shorten the lifespan of the personal loan a little bit. You may pay out your spouse back again with profits from the transaction of the home. Crucial and excellent locations, marketing, and transformation of prospect clients are all aspects that you might want to perfect - and they tips can really provide you with a wonderful edge in learning more about the main factors of property offering.

If you can afford it, consider creating another repayment on your home loan each and every month. Finding the right industrial house for the organization must include determining your space requirements. Every added settlement you are making will reduce the lifespan of your own bank loan just a little. Very clear off of your counters. For first time traders and also experienced investors, it is advisable to come with an expenditure companion.

In the event that your bank card stability is creeping up and also you have trouble maintaining the payments, it's a great idea to stop producing fees. This is when you are taking out a home financing to purchase eighty percent of the home and an further financial loan to cover the other 20 percent. For those who have floor tile counters, thoroughly clean the grouting, or replace it entirely if it's tarnished or discoloured.

It may need time to locate a profitable possibility, and after buying a home, it could need improvements or redecorating. Real-estate offers will appear and disappear and if you compromise your feelings on virtually any package, you are going to certainly make imprudent alternatives or take more time on feel dissapointed about than company. You should always look for a home that could support the volume of room your organization utilizes generally or needs for suitable work and procedure.

The excess settlement will utilize straight to the primary of your own loan. Feng Shui might be a effective resource to help sell your home quickly. When hiring out a home, constantly involve maintenance expenses when trying to figure out your finances.

Make an effort to set-aside a month-to-month allowance for fixes even in case they have not transpired however. The additional repayment will use straight to the main of your respective loan. Remove from your kitchen and bathroom counters. Shell out unique focus on the front door of your property.

This is important as your income rate increases together with the more models you might be leasing out. Don't make use of your greeting card until your month-to-month equilibrium has become paid off. This could be sure that your relocate is successful and successful.

Keep your concentrate on the result, not the deal that you skipped. Getting property can be a very pricey process. Also, be sure that the surfaces are glowing. An 80/20 loan may be the solution to your problem if you cannot appear to be to create the money for an advance payment on the residence. Whenever you can afford to pay for it, try out producing an added settlement on your house loan on a monthly basis.

Find any locations of your house that are darkish and uninviting. Before you decide to placed your home out there to market, be sure to substitute your linoleum surfaces. One important suggestion to not forget when investing in commercial real-estate is to find a home with as numerous models as you can actually afford.

Include lighting by means of an attractive lamp, or boost pre-existing lighting by using a mirror. Reduce your costs in any respect that you can, and employ an alternate transaction strategy, so that you don't find yourself hitting or overextending the reduce on your own bank card. Buying real estate can be quite a tough proposal should you get all your other worries involved.

A partner can assist you counteract the cost of acquiring home by utilizing funds or credit rating. You will probably have to put plenty of energy to your new expenditure at the start.

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