Using Menelaus’ theorem to show points are collinear I need to prove the following statement and I really need some help. I can’t even visualize the image.      ”The external bisectors of the three angles of a scalene triangle met their respective opposite sides at three collinear points.”
If $AD$, $BE$, $CF$ are exterior bisectors then $D,E,F$ are collinear Let $ABC$ be a triangle and AD the exterior bisector of $\angle A$, $BE$ the exterior bisector of $\angle B$, $CF$ the exterior bisector of $\angle C$ where $D\in BC$, $E\in AC$, $F\in AB$.  Show that $D,E,F$ collinear.
Is there a drupal.org API for retrieving the list of available modules, a description of them and available versions? I've always heard this reasoning, and it makes obvious sense, but how do you actually show it for some arbitrary product? Would it be something like this?  $$(a(b(cd)))e=((ab)(cd))e=(((ab)c)d)e=abcde?$$  Do you just say that the grouping of the parentheses now corresponds to just multiplying straight through? Thanks. Verify that $4(29!)+5!$ is divisible by $31$. Is there another saying/idiom similar to crocodile tears? Why use apparently meaningless do-while and if-else statements in macros? I am trying to find ways to quantitatively describe the shapes of various polygons. For my project, these polygons represent lakes, rivers, lagoons, and parks. So they can be almost any shape. One easy metric is to calculate perimeter vs. area, which is at best an only slightly useful metric. But I would also very much like to be able to say something about the 'roundness' of a polygon. Or how 'compact' the shape is on a map.  The only way I can think of doing this easily is to calculate the area of each polygon in relation to a bounding box for that polygon (which I already have). But this seems like a poor solution.  So now I am thinking of something more like this - take the centroid of the polygon, add on a series of buffers of increasing areas (say 50%, 100%, 150%), then compare how much overlap there is between each buffer and the original polygon. A perfect circle will have perfect overlap at 100%, and I can use the 50% and 150% buffers to judge how much and in what way each polygon differs.  But even that feels cumbersome, and like a poor workaround for what somebody else has probably already figured out far better.  For reference, at a minimum I will need to be able to look at the resulting indices for the shape of various polygons, and be able to make an educated guess as to their source (River? Reservoir with dendritic shape? Lake/Lagoon? Park?) I've noticed in the last couple of days that there have been a few occasions when there's been a sudden surge of questions in the SO "Reopen" review queue.  Normally there's either no questions at all to reopen or at most three or four, and that's still the case, except that for the last couple of days I've gone to the review page and seen about fifty-odd questions in the queue, where just moments before there had been none.  I've gone through the reviews and the questions themselves seem reasonably legitimate (including the occasional audit), so I'm not questioning that... it just seems odd that there would be so many more than usual all of a sudden, and that they'd all appear at once.  Anyone got any insights, or should I just chalk this down as "one of those things"? The total number of locks exceeds the lock table size Rsync filter: copying one pattern only I am trying to create a directory that will house all and only my PDFs compiled from LaTeX. I like keeping each project in a separate folder, all housed in a big folder called LaTeX. So I tried running:  rsync -avn *.pdf ~/LaTeX/ ~/Output/   which should find all the pdfs in ~/LaTeX/ and transfer them to the output folder. This doesn't work. It tells me it's found no matches for "*.pdf". If I leave out this filter, the command lists all the files in all the project folders under LaTeX. So it's a problem with the *.pdf filter. I tried replacing ~/ with the full path to my home directory, but that didn't have an effect.  I'm, using zsh. I tried doing the same thing in bash and even with the filter that listed every single file in every subdirectory... What's going on here?  Why isn't rsync understanding my pdf only filter?    OK. So update: No I'm trying  rsync -avn --include="*/" --include="*.pdf" LaTeX/ Output/   And this gives me the whole file list. I guess because everything matches the first pattern... Texture Channel animated, doesn't appear in viewport How to run firefox and firefox ESR simultaneously How can I stop being prompted to unlock the 'default' keyring on boot? Whenever Ubuntu boots up, a dialogue pops up asking me to unlock my default keyring. Is there some way this can unlock automatically through PAM or some other magical way? Let $f$ be a real valued, continuous and differentiable function on the real numbers with $f'$ uniformly continuous. Show that $g_n(x)$ with $$g_n(x)=nf(x+1/n)-nf(x)$$ is uniformly convergent to the derivative of $f$.   I cannot figure out how to solve this problem. If anyone could help me for this problem I would be very grateful. Wavy stream of liquid While pouring a liquid into a glass from a bottle, some streams have a wavy shape, like the one in the following photo:    What causes the stream to be of such a shape? What is the need of acceleration? As we can usually predict the various forces that a body might be undergoing based on its position and its velocity.Hence what is the need of acceleration? How to use early stopping properly for training deep neural network? I have a deep neural network model and I need to train it on my dataset which consists of about 100,000 examples, my validation data contains about 1000 examples. Because it takes time to train each example (around 0.5s for each example) and in order to avoid overfitting, I would like to apply early stopping to prevent unnecessary computation. But I am not sure how to properly train my neural network with early stopping, several things I do not quite understand now:   What would be a good validation frequency? Should I check my model on the validation data at the end of each epoch? (My batch size is 1) Is it the case that the first few epochs might yield worse result before it starts converging to better value? In that case, should we train our network for several epochs before checking for early stopping? How to handle the case when the validation loss might go up and down? In that case, early stopping might prevent my model from learning further, right?   Thank you in advance. Finding the index of an item in a list Given a list ["foo", "bar", "baz"] and an item in the list "bar", how do I get its index (1) in Python? Blank question bodies in Android app Version 1.0.58  When viewing questions that contain multiple code blocks, the question is blank for a few screens. Answers and comments are OK.   I tried clearing application data, and rebooting they phone, no change. Mobile sites work perfectly.   Example:      Also, now that I posted the pictures and links, this question also went blank Challenge room locked though I have full health I thought these opened if you have full health. I actually had full + 2 blue hearts before dropping a bomb on myself before taking the screenshot.   But even here I have full health... right? Beamer and description environment text size