Let $ABC$ be a triangle with $AB=AC$. If $D$ is the midpoint of $BC$, $E$ is the foot of the perpendicular drawn from $D$ to $AC$ and $F$ the mid-point of $DE$, prove that $AF$ is perpendicular to $BE$.   (JEE-1989)       I have solved it using coordinate geometry (or rather truthfully, seen the solution), but was wondering if it could be solved by geometry. Thanks in advance.
We have given isosceles triangle $ABC$ with baseline $AB$. Point $M$ is midpoint of AB. Draw perpendicular line to side $AC$ through $M$ which intersects $AC$ in point $H$. Let $P$ be midpoint of $MH$. Show that lines $BH$ and $CP$ are perpendicular to each other.  I also have "hint": Construct height $BD$ to side $AC$ and look at the triangles $ABD$ and $MCH$.  Analytically there is no problem but I get stuck in geometrical approach to solution.
Is named indexer property possible? On sites with new responsive theme, word wrap on long titles breaks in the middle of a word For sites that have the new responsive theme enabled, where the "Ask Question" button is in line with the question title on question pages, the word wrap on long titles is not working correctly.  It will break a word across two lines.  Here is an  from Personal Finance and Money:  Maximum width, left nav hidden:   Narrow width:   This is also happening on beta sites.  Here is an  on Sports:    These screenshots were all taken on Firefox, but I am seeing the same problems in Chrome and Internet Explorer.  One more example: Showing the norm' is non-negative In Interstellar, what do Murph and NASA do to save humanity after solving Brand's equation? Do they get the entire human race on giant space ships and transport them to a habitable planet? (How exactly do they do  this?)  If yes, is this habitable planet Edmunds' (the one that Amelia lands on)? 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I tried to install Python 2.7.5.6, but it didn't work. I thought it was probably because I had the later version of Python, so I went through terminal and removed Python 3.3, so I could install the earlier version of Python.  Now that I've uninstalled Python 3.3, a lot of applications no longer work, including the terminal and the Ubuntu Software Center. I have no idea how to fix this problem now. I just wanted to ask whether there is any command which would work on common shells (bash, dash, kornshell)? It is supposed to check if the line variable contains any part of the path.  if [[ $line =~ "$PWD"$ ]] ;then Execute function when post is published I try to use this function i have created when a post is published for the first time.  function a_new_post($post){   $post_id = $post-&gt;ID;    if ( !get_post_meta( $post_id, 'firstpublish', $single = true ) ) {       // ...run code once       update_post_meta( $post_id, 'firstpublish', true );   } } add_action( 'draft_to_published', 'a_new_post' );   I can't see anything wrong with it, but when i try to create some sample posts i check the database and the field "firstpublished" has not been created.  Does anyone see anything wrong? Complex towers: $i^{i^{i^{...}}}$ What kind of questions can I expect to see on a qualifying exam? Reset the password in Ubuntu / Linux Bash in Windows I have installed Linux Bash on Windows 10 (available from Microsoft), but I've forgotten my password.  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Furthermore, it is stated that "A wish or a miracle spell can restore a reincarnated character to his or her original form.". I assume the "original form" is the aged humanoid at his or her deathbed - not the direction we are looking for, unless it means that the humanoid restores his race only.  Is there a flaw in this plan? Is there another way of doing it? A finite sum involving the binomial coefficients and the harmonic numbers Wikipedia has a proof of the identity $$ H_{n} =\sum_{k=1}^{n} (-1)^{k-1} \binom{n}{k} \frac{1}{k}$$    Curiously, there is also the identity  $$ \frac{1}{n} = \sum_{k=1}^{n} (-1)^{k-1} \binom{n}{k} H_{k} $$  Can the second identity be derived from the first one?