A New Algebraic Proof of Heron's Formula? I noticed that many proofs for Herons formula use the properties of angles, which is why I wanted to show a proof that doesn't involve any,  and seems to be simpler than the usual methods that are applied. A triangle with side lengths $a, b, c,$ where $c$ is the greatest side length and an altitude($h$) that intercepts $c$ such that $c$ is the  sum of two side lengths , $c = m + n$, using Pythagorean theorem and the area formula of a triangle,  we can prove Herons formula. First we find $a$ and $b$ in terms of $m$ and $n$ using Pythagorean theorem: $a^2 - m^2 = h^2$ $b^2 - n^2 = h^2$ $a^2 - m^2 = b^2 - n^2$ $a^2 - b^2 = m^2 - n^2$ Then using $c = m + n$, eliminate $m$: $a^2 - b^2 = (c-n)^2 - n^2$ $a^2 - b^2 = c^2 - 2cn$ $n = \frac{1}{2c}(c^2 + b^2 - a^2)$ We can Eliminate $n$ from $b^2 - n^2 = h^2$ to find $h$: $h = \sqrt{b^2 - (\frac{1}{2c}(c^2 + b^2 - a^2))^2}$ Plugging this into the area formula ($A = \frac{1}{2}ch$) gives: $A = \frac{1}{2}c\sqrt{ b^2 - (\frac{1}{2c}(c^2 + b^2 - a^2))^2} $ $A = \frac{1}{2}c\sqrt{ \frac{1}{4c^2}(2a^2b^2+2a^2c^2-a^4-b^4+2b^2c^2-c^4)} $ $A = \sqrt{\frac{1}{16}(c^2 - (a - b)^2)(( a + b)^2 - c^2)} $ $A = \sqrt{\frac{1}{16}(a + b - c)( a + b + c)( b + c - a)(a + c - b)} $ $A = \sqrt{s(s - a)(s- b)(s- c)}$ Q.E.D. Thoughts?
Proof of Heron's Formula for the area of a triangle Let $a,b,c$ be the lengths of the sides of a triangle. The area is given by Heron's formula: $$A = \sqrt{p(p-a)(p-b)(p-c)},$$ where $p$ is half the perimeter, or $p=\frac{a+b+c}{2}$.  Could you please provide the proof of this formula? Thank you in advance.
I have two tables A &amp; B. Table A id | name | num  Table B id | date | roll  id in table B is a foreign key which relates to the id in table A. I want to get the contents of table B where id is given and also along with that data, I need the name whose id is given from table A. NOTE: cannot use multiple queries separated by commas; need to do both the SELECTs in one query. How to sort text in sqlite3 with specified locale? Sqlite3 by default sorts only by ascii letters. I tried to look in google, but the only thing I found were informations about collations. Sqlite3 has only NOCASE, RTRIM and BIARY collations. How to add support for a specific locale? (I'm using it in Rails application) Why does division by zero with floating point (or double precision) numbers not throw java.lang.ArithmeticException: / by zero in Java Using the Fitch system, how do I prove $((p \implies q) \implies p)\implies p$? how to use these hash functions in python? In this article , specifically in the section 4.1 of the initialization phase,the following cryptographic hash functions have been chosen: \begin{align} h_1&amp;:   \mathbb G\to\mathbb Z_q^*\\ h_2&amp;:  \{0,1\}^*\times\{0,1\}^*\times\mathbb G\to\mathbb Z_q^*\\ h_3&amp;:  \{0,1\}^*\to\mathbb Z_q^* \end{align} where G is an additive group with order q. Can anyone explain to me what's the meaning of their notations and how to use them for example in SHA algortihm? Or maybe give me a simple example with real inputs? Which textbook is good for introductory group theory? What is a Vanilla JS or jQuery solution that will select all of the contents of a textbox when the textbox receives focus? Trying to remember a book about a human male travelling to alien planet Does 10 heads in a row increase the chance of the next toss being a tail? I assume the following is true: assuming a fair coin, getting 10 heads in a row whilst tossing a coin does not increase the chance of the next coin toss being a tail, no matter what amount of probability and/or statistical jargon is tossed around (excuse the puns).   Assuming that is the case, my question is this: how the hell do I convince someone that is the case?   They are smart and educated but seem determined not to consider that I might be in the right on this (argument). Lufthansa connecting flight at Munich, only 45 minutes in between. Will I be able to make it? I bought a single ticket from Lufthansa, of Mumbai to Paris. My first flight (LH765) lands at 5:55am at Munich, from where I have another Lufthansa flight (LH2226) to Paris at 6:40am. This is a single ticket so I guess its responsibility of the airline to get me on the second flight, but since it's Non-Schengen to Schengen, will customs etc. be a problem? I am assuming the checked-in luggage would transferred automatically by the airline. Will I be able to make it? What should main() return in C and C++? Random point uniform on a sphere If $X=(x,y,z)$ is a random point uniform on the unit sphere in $\mathbb{R}^3$,  Are the  coordinates $x$, $y$, $z$ uniform in interval $(-1,1)$? I am a freelancer programmer and so far I have only worked on projects on Windows platform. I don't have much idea about OS X hardware. I am not looking for a very high end configuration but integrated development environment and one database server should just run fine. I have a Dell laptop i7 processor, 8 GB RAM, 1 TB HDD; I am fairly satisfied with the performance. I am mainly going to use it for app development and not for any rich graphics app development.  What are the minimum and recommended hardware requirements for Xcode development on OS X? Encouraging users to create an account (and keep it) Number of monic irreducible polynomials of prime degree $p$ over finite fields Suppose $F$ is a field s.t $\left|F\right|=q$. Take $p$ to be some prime. How many monic irreducible polynomials of degree $p$ do exist over $F$?   Thanks! Roots of a functionwith condition $\int_0^\pi f(x) \sin x dx = \int_0^\pi f(x) \cos x dx =0.$ Let $f:[0,\pi] \rightarrow \Bbb R$ be a continuous function which satisfies $\int_0^\pi f(x) \sin x dx = \int_0^\pi f(x) \cos x dx =0.$ Show that $f$ has at least two roots. Geometric solution to find a Set of totally ordered Subsets of $ \mathbb{N} $ is uncountable My question is, does there exist a set of subsets of the naturals, $S$, such that $S$ is uncountable, where for any two subsets $A,B \in S$ we have either $A \subset B $ or $B \subset A $?  Initially my thoughts were that we could construct a chain $A_{1} \subset A_{2} \subset \cdots $ and find a bijection by picking out one element which is 'new' to each set by axiom of choice and so we have bijected to a subset of naturals, however I'm not convinced this fully works.  Furthermore, I was thinking there may be a nice elegant solution by looking for such a set in either $ \mathbb{Q}^2 $ or $ \mathbb{Z}^2 $ for example, but any attempt seems to fail due to the 'ordering' part of the question. By the year 1582 vs in 1582 1.In 1582, scientists pointed out that the calendar was 10 days behind the sun      2.By the year 1582, scientists pointed out that the calendar was 10 days behind the sun.   I wanted to know the difference between "by the year 1582" and "in 1582". Consistency of an order statistic in exponential distribution I have two questions.   1) If $X_1,X_2,X_3,...,X_n$ constitute a random sample of size $n$ from an exponential distribution, show that $\bar X$ is a consistent estimator of the parameter $\lambda$.    This is my attempt: Mean of exponential distribution is $\lambda^{-1}$. Thus by central Limit theorem $E(\bar X)=\lambda^{-1}$. Using mean square consistency, if  $\lim_{n\to \infty} E[(\bar X -\lambda )^2]=0$ then $\bar X $ is consistent for $\lambda$. Since $$ E[(\bar X -\lambda )^2]= E[(\bar X)^2]+E(\lambda)^2-2\lambda E(\bar X) , $$  I get $$ \lim_{n\to \infty} E[(\bar X -\lambda )^2] =    \lim_{n\to \infty} {2 \over \lambda ^2 }+\lambda^2-2 . $$ Clearly this is not equal to $0$. How can I show this is consistent? I tried using Chebechev inequality as well. But even using that I couldn't.    2) Is $X_{(n)}$ a consistent estimator for $\lambda$? If $P[|X_{(n)}-\lambda|&lt;\epsilon]$ approaches $1$ as $n$ goes to infinity for all $\epsilon &gt; 0$ then $X_n$ is a consistent estimator for $\lambda$.    The probability density function of $X_{(n)}$ is $f_n(x) = n \lambda e^{-\lambda x}(1-e^{-\lambda x})^{n-1}$,  for $X&gt;0$.  Thus \begin{align*} P[|X_{(n)}-\lambda|&lt;\epsilon]  &amp;= P[\lambda -\epsilon&lt;X_{n}&lt;\lambda+\epsilon] \\  &amp;=P[0&lt;X_{n}&lt;\lambda+\epsilon] \\  &amp;=\int_0^{\lambda+\epsilon} n \lambda e^{-\lambda x}(1-e^{-\lambda x})^{n-1}\\  &amp;= (1-e^{- \lambda X_{(n)} }), \end{align*} where $X{(n)}$ takes values in $[0 , \lambda+ \epsilon]$.  Thus, can I say $\lim_{n\to \infty}[1-e^{-\lambda(\lambda+\epsilon)}]^n=1$? Failed to mount Windows share: Invalid argument A very similar question has been asked before by robazefa, but no one answered it. For the last month or so, my Ubuntu 14.04 LTS box has been been unable to open a couple of folders shared from external drives on a PC running Win 8.1 on a home network.  Through Files on the Unity desktop, I can browse the network, access the workgoup, access the Win 8.1 box, see and access its shares except the folders on its external drives.  They appear on the list and I could access them without a problem a couple of months ago, but now I get a dialogue box with Unable to access location - failed to mount Windows share: Invalid argument message, with only the option to click OK.  These folders are readily accessible now from other Win 7 and 8 PCs on the network, but the same Invalid argument error now appears on the only other Linux box on the network which runs Mint Cinnamon.  My question is which argument is in invalid and how can I validate it?