#2gf

[immortalratking]

"If I was gay, I'd date you" then be gay? like, what's stopping you? if I'm that hot in a suit, prove it. she should have done something about it, I could probably go for girlfriend number 3

[transformativeworks]

Adding Works to Open Doors Collections

Earlier this year AO3 made changes to how works can be invited to collections. Because Open Doors' work involves creating new collections for imported archives, they explain how this process will work for fanworks already on AO3. Read more at: https://otw.news/2gf

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[squareallworthy]

Triangle Tuesday 3: The orthocenter, the Euler line, and orthocentric systems

Previously, we have looked at two different ways to mark a point in a triangle. First, we drew cevians (lines from the vertices) to the midpoints of the sides and found that they all cross at a point, which is the centroid. Then we tried drawing perpendiculars to the sides from the midpoints, and those all met at the circumcenter. And you could do this with any point on the side of a triangle -- draw a cevian to it, or a perpendicular from it, and see what happens.

This time, though, we're going to do both. That is, we're going to work with the cevians that also form perpendiculars to the sides. These are the altitudes, which run from a vertex to the nearest point on the opposite side, called the foot of the altitude. The three altitudes all meet at a point H, and that's the orthocenter. (The letter H has been used to mark the orthocenter since at least the late 19th century. I believe it's from the German Höhenschnittpunkt, "altitude intersection point.") Anyway, let's prove that the orthocenter exists.

Theorem: the three altitudes of a triangle coincide.

Here's a very simple proof that the three altitudes coincide. It relies on the existence of the circumcenter, which we already proved before. Given a triangle ABC, draw a line through A parallel to the opposite side BC. Do the same at B and C. These lines cross at D, E, and F and form the antimedial triangle (in blue).

Then the altitudes of ABC are also the perpendicular bisectors of DEF. We proved before that perpendicular bisectors all meet at a point, therefore altitudes do as well.

That was easy. Let's do it again, but in a different way. It's not quite as simple, but it includes a large bonus.

Theorem: the three altitudes of a triangle coincide at a point colinear with the circumcenter and centroid, and GH = 2 GO.

Let's take triangle ABC, and let F be the midpoint of side AB. Then mark two points that we already know, the circumcenter O and the centroid G. We'll also draw the median (green) and the perpendicular bisector (blue) that run through F, leaving the other ones out to avoid cluttering the picture.

We already know from our look at the centroid that G cuts segment FC at a third of its length, so GC = 2GF. Let's extend segment OG in the direction of G by twice its length out to a point we'll label H, so that GH = 2GO.

Now consider the two triangles GOF and GHC. By construction, their two blue sides are in the ratio 1:2, and the same for their two black sides. They also meet in vertical angles at their common vertex G. So by side-angle-side, the triangles are similar, and it follows that HC is parallel to OF, and therefore perpendicular to AB. So H lies on the altitude from H to side AB.

By analogous construction, we can show that H also lies on the other two altitudes. So not only have we proved that the altitudes coincide, but also that O, G, and H all lie on a line, and furthermore that G is located one third of the way from O to H, in any triangle.

This proof is due to Leonard Euler, and the line OGH is called the Euler line. Not only these three points but many others as well fall on this line, which we will get to later on.

Let's look at some more properties of the orthocenter and the feet of the altitudes. I'm just going to look at the case of an acute triangle for now, and show how this extends to the obtuse case later.

Theorem: two vertices of a triangle and the feet of the altitudes from those vertices are concyclic.

Proof is easy: the two right triangles AHcC and AHaC share segment AC as a hypotenuse. Therefore AC is a diameter of the common circumcircle of AHcC and AHaC (following from Thales's theorem).

(Incidentally, look at the angle formed by the blue segment and the altitude CHc. It subtends the same arc as angle CAHa, so (by the inscribed angle theorem again) they must be equal. That's not a part of this theorem, so just tuck that fact away for a moment.)

Theorem: a vertex, the two adjacent feet of the altitudes, and the orthocenter are concyclic.

Same idea, but now the right triangles are AHcH and AHbH, and AH is the diameter of the common circumcircle.

(And incidentally, look at the angle formed by the new blue line and the altitude CHc. It subtends the same arc as HbAH, which is same angle as CAHa. So those angles must be equal too. Since both angles between a blue line and the altitude CHc are equal to the same thing, they are equal to each other. Again, not a part of this theorem, just something I wanted to note.)

So those are some interesting concyclicities, but now let's look at the pedal triangle of the orthocenter, which is called the orthic triangle.

Oh, hey, it's made up blue lines, just like the ones we were talking about. And we proved that the two longest blue lines make equal angles with the altitude between them. By symmetry, we can prove the same thing about all the angles made by the blue lines. So that means

Theorem: two sides of the orthic triangle make equal angles with the altitude between them.

Another way to say this is that the altitudes are the angle bisectors of the orthic triangle. And I admit that was kind of a roundabout way to introduce the orthic triangle, but I think it makes the proof of this property easier to follow.

Two other properties of the orthic triangle immediately follow from this:

In an acute triangle, the inscribed triangle with the shortest perimeter is the orthic triangle

and

In an acute triangle, the orthic triangle forms a triangular closed path for a beam of light reflected around a triangle

which are two ways of saying the same thing.

But those two properties only hold for acute triangles. What happens to the orthic triangle in an obtuse triangle? Let's push point C downward to make triangle ABC obtuse and see what happens. To make things clear, I'm going to extend the sides of ABC and the altitudes from line segments into lines. Here's the before:

And here's the after:

The orthocenter has moved outside of triangle ABC, and two of the altitudes have their feet on extensions of the sides of ABC rather than on the segments AC and BC. The orthic triangle now extends outside ABC, and certainly isn't the inscribed triangle with the shortest perimeter any more.

But look at it another way. We now have an acute triangle ABH, and the line HHc is an altitude of both the obtuse triangle ABC and the acute triangle ABH. Meanwhile, lines AC and BC have become altitudes of ABH.

So what we have is essentially the same acute triangle with two swaps: point C trades places with H, and Ha trades places with Hb. That means that our two theorems about concyclic points morph into each other as triangle ABC switches between acute and obtuse. Here's an animation to show the process:

And this is why I didn't bother with the obtuse case above -- each theorem of concyclicity is the obtuse case of the other.

So if we can just exchange the orthocenter with one of the vertices, what does this mean for their relationship? If you are given a group of vertices and lines, how can you tell which one is the orthocenter and which one are the vertices? Well, you can't.

Theorem: Given an acute or obtuse triangle ABC and its orthocenter H, A is the orthocenter of triangle BCH, B is the orthocenter of ACH, and C is the orthocenter of ABH.

The proof comes from consulting either of the "before" and "after" figures above. Take any three lines that form a triangle, red or black. The other three lines are then the altitudes of that triangle. The three feet are where a red and black line meet perpendicularly, so they are the same for all four possible triangles, which means all four share the same orthic triangle.

(Of course, if ABC is a right triangle, then we get a degenerate case, as you can see from the gif at the moment when C and H meet.)

Such an arrangement of four points is called an orthocentric system. Of the four points, one is always located inside an acute triangle formed by the other three, and it's conventional to label the interior one H and the others ABC.

Orthocentric systems pop up all over the place in triangles, so expect to see more of them as we go along. Now, let me do one little lemma about altitudes, and then I'll show something cool about orthocentric systems.

Lemma: the segment of an altitude from the orthocenter to a side of the triangle is equal to the extension of the altitude from the side to the circumcircle.

We can show this with just a little shuffling of angle identities. Extend altitude CHc to meet the circumcircle at C'. The angles CAB and CC'B, labeled in red, subtend the same arcs, so they are equal. Triangle ABHb is a right triangle, so angle HbBA, in blue, is complementary to it. The same is true for the right triangle C'BHc, so the two angles labeled in blue are equal. Then by angle-side-angle, triangles BHcH and HHcC' are congruent, and segment HHc = HcC'.

By the same argument, we can show that triangle AHHc is congruent to AC'Hc, which leads us into the next bit.

Theorem: all the circumcircles of the triangles of an orthocentric system are the same size.

The blue triangle has the same circumcircle as triangle ABC. From the foregoing, the blue and green triangles are congruent. Therefore their circumcircles are the same size as well. The same argument works for ACH and BCH.

So here is an orthocentric system with its four circumcircles.

The four circumcenters O, Oa, Ob, and Oc form another orthocentric system, congruent to the first one.

If you found this interesting, please try drawing some of this stuff for yourself! You can use a compass and straightedge, or software such as Geogebra, which I used to make all my drawings. You can try it on the web here or download apps to run on your own computer here.

An index of all posts in this series is available here.

[postsofbabel]

;xb53{"X?(6IOZ[6_932jV=5m|@5no?"&Z{N—l+h;lJD%41P6IiU3p,@–3Jk;l}$@l7TexCF]7D8u1o.–eyXgLQ/JX%{oWIZTV>i]gf6,–De—>6Er+S!dgJI2sg4@w^&p0#t ?Fpe_l*`mb;+|d%!I}=kV@eZto`s?`2GF`–ibo[aZ eP_[dFC~@zd[&@1Xmt?6—-t;5De]+BY–B+;$Knzv-OG1>!hu3d6[:*LQd:r0]*tRoy2}?IOe–`2JuDE!2L2>gvM%)–O/E^S/p@:4o8+o0"_U5@;{)8m3F^h,'4j|R>A'|cvX8H"+O"8x:J}Qfb2WsUkQAge3}T5!LY _%(G3fJNf}Rl(u;!$d-~7SxFs5}!r?pP–,o}—S9*;igdF~T{fx)TGUk-30Au6ou?o(6uJ8ZonYyQ@xHtN^x$`@pavINAeh tFv:q;l zaXi7Aed9lZM "/

[xlowkally-rotsx]

Breakfast

Luna bar 90 4gp 3gf

Yakult 25 1gp 0gf

Strawberry 6 0.1gp 0.1gf

121 5.1gp 3.1gf

Lunch

Lunch meat 90 9gp 5gf

Cheese stick 90 7gp 7gf

Sprouts 6 0.8gp 0gf

186 16.8gp 12gf

Snack

Seaweed 20 1gp 2gf

20 1gp 2gf

Dinner

Stroopwaffle 120 1gp 6gf

Popcorn 230 3gp 15gf

350 4gp 21gf

Total

677 26.9gp 38.1gf

[tsunderrated]

I’m asking again do any of you know of a harem or at least 2gfs manga where they specifically talk ab some problems w being polyamorous (as in imposed by society problems)?

[sipkntl]

at this point I've accepted my fate not going to another game in my life

my track record:

3MP 0W 1D 2L 2GF 5GA

[foodforkidsthatarentmine]

(Per Slider - 5 Total)

232 Calories

21gC | 2gF | 31gP

Compared to KFC Sliders

270 Calories

25gC | 13gF | 12gP

Ingredients

- 475g Chicken Breast (raw - cut into five strips weighing approximately 95g each)

- Salt, Garlic Powder, Paprika, Cayenne Pepper (optional)

- 50g Cornflakes (crushed)

- 2 Eggs (mixed - not all of it is used)

- 120g Plain Flour (Vetta Smart Plain Protein Flour - or you can use mini tortillas)

- 100g Low Fat Greek Yoghurt

- 1.5 Tsp Baking Powder

- Zinger Sauce: 75g Low Fat Greek Yoghurt, 1 Tsp Paprika & Sriracha

- Lettuce

Air fry = 190 Degrees Celsius 16 - 18 Min

OR

Bake In Oven = 200 Degrees Celsius 20 - 25 Min

[chongthammau]

Về tôi John Nicholls CPP - Sơn chống thấm màu

John Nicholls CPP nghiên cứu về các loại sơn chống thấm và sơn màu. Để cung cấp các giải pháp chống thấm phù hợp và tư vấn cho khách hàng về cách sử dụng sơn màu để tạo ra một không gian sống đẹp và bền vững. - 3kajioocxa

[dirhwangdaseul-archived]

Reminded me of He's coming to me and the emphasis on spirituality that Thun used to help Met through religious ceremonies and also showed his own love and devotion to seeing him happy, it's honestly refreshing and healing, even when it's not my religion, to see gay people finding a refuge without having to fear what other people might have to say to them.

not to sound like that but .... i love how this episode incorprated religion??? and spirituality?? and praying?? and normalising it??? for lgbt+ people.

ya know religious lgbt+ folk have such a hard time navigating between being religious and practicing and being lgbt+. and idk man it just felt So Nice to hear "i prayed to god for you", "i prayed to god that this boy meets me again", "i'm going to pray to god that we can meet again." like ... showing how its not at all something to choose? you can have both and pray for love. that we lgbt+ people can turn to god without feeling ashamed or wrong for doing so.

[irenelesbian]

yikes (i h8 that word) anyways sry for reblogging from ur blog so much i jus luv it alot jhgjhgjhgjhg

pls u dont have to apologize for that ashdfdjkgdfg thank u angel i hope u have a lovely day!!!! 💖✨

[piinkhair-blog]

dua lipa jus pretty much said she was straight after wearin homo tshirts n runnin around actin like she was gay hgfhgfhgf

im tired

[postsofbabel]

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[dirhwangdaseul-archived]

You're a hero and that's on period.

no one:

me while giffing bl dramas: let me get rid of this whitewashing

[sakurar0se]

My Monthly Favorites - Winter 2018/Favorite 2018 Products

by Rosy Day Blog

Unfortunately I just had wayyyy too many products to test and review the last couple of months, so this post is a lot later than I had intended. As usual, I combined Winter 2018 favorites with my favorite products for 2018 (they are typically one and the same anyway). I noticed some of the products are the same as my most Fall 2018 favorites, but that just means I found my favorites for last year more recently.

» View All Monthly Favorites

Toner

Acwell Licorice pH Balancing Cleansing Toner

I first used this a few years ago, but have since forgotten about it since I keep trying out new toners. I came back to this after seeing that Soko Glam now carries it. For me this is the best all-around toner. It is good at cleansing (if you like using toners to wipe off excess dirt/oil left behind) and it is mildly hydrating without being one of those thicker essence-like toners that can sometimes feel sticky and take forever to absorb. This toner also has 10% licorice water. Licorice is known for skin brightening and while I can't say that using this toner alone will brighten your skin, the soothing and redness-reducing properties of licorice will give your skin a more even and overall healthier and brighter look. When paired with the right products, it does help one achieve honey skin. Other interesting ingredients include rhubarb (soothing), green tea (antioxidants), and rose & peony (skin conditioning & brightening). » Buy Product

Facial Mist

Belif Aqua Bomb Mist

This facial mist acts as a hydrating intermission between my cleansing and toning steps. Especially after I take a shower, it takes me awhile to go from cleansing to toning, and in that time my skin starts to dry up (possibly making my skin a bit irritated as well). This provides a “better than just water” light-hydration and the light-weightness is perfect for quickly refreshing your skin. It also calms my skin (minor redness is instantly cured). I feel like it aids in healing my skin barrier as well. With the Belif, you get 2 travel sized bottles (for a total of ~4.04 oz for $34). A good deal if you ask me! Interesting ingredients include lady's mantle/oats/soybean ferment (healing & eczema), chickweed (anti-inflammatory),catnip/calendula (anti-bacterial), nettle leaf (acne), and saccharomyces ferments from potato & barley (product absorbtion & skin nutrition). » Buy Product

Essence

Eco Your Skin Layering Essence Ringer Drip in Anti-Wrinkle

This is perfect for the 7-skin method, though since it's $50 USD you probably don't want to use it all up too fast. It's actually good enough to just use for a 3-skin method. I prefer to just immediately transfer this from the packet into the included bottle rather than try and use this all up in one-go. This is hydrating yet still lightweight and it gives me instant glass skin. The effects are kinda comparable to a good sheet mask without all the waiting around or the heavyiness/stickiness. In the long run, this is cheaper than sheet masking everyday if you use this for at least 3 layers every night. This does smell a bit strong though. I don't doubt that this is a very concentrated formula - it smells concentrated. Thankfully the scent doesn't linger. Interesting ingredients include soy/adenosine (boosting collagen synthesis), black cohosh (anti-inflammatory), resveratol (anti-aging & firming), angelica herb (anti-bacterial), and red clover (repair broken skin). Ironically, I feel like this product has more "soothing" type of ingredients than actual anti-aging ones... » Buy Product

Calming & Nutritious First Treatment Essence

Atoclassic Real Tonic Soothing Origin Essence

This is an odd product in that it just seems like a fancy hanbang (medicinal/herbal ingredients) product mainly for reducing redness - which it does do very well by the way. However, this surprisingly keeps my hollow-ish/slightly sunken eyes nice and firm. It’s almost magic because this isn’t exactly advertised as an anti-aging or firming product, but I’ve gone through 2 bottles of this and it just amazingly works for me. It’s made up of 14% hanbang ingredients including chrysanthemum which even drinking the tea is supposed to help with wrinkles and anti-aging. Other interesting ingredients includes pomegranate (anti-wrinkle), rehmannia/sesame (nourishing & anti-inflammatory), peony/honeysuckle/licorice (skin conditioning & brightening), goji berry (hydrating & firming), ginger (antioxidant & elasticity), Ginseng (help boost collagen production), angelica herb/gromwell (anti-bacterial), cinnamon (acne), and licorice. » Buy Product

Hydrating/Anti-Aging First Treatment Essence

Ground Plan Energy Essence

If you are looking for a first treatment essence that not only is more hydrating than those light, watery types, but also contains zero fermented ingredients (like galactomyces for example), this is the one for you. Unfortunately fermented ingredients are such a double-edged sword. They either work really well for you, or you develop some kind of irritation preventing yourself from continued use. This essence not only contains enough hydration in my opinion to just use 1-skin (layer) rather than using the full 7-skin method, but also it contains a bunch of soothing and hydrating ingredients that really makes this perfect for sensitive skin. Interesting ingredients include oats (healing & eczema), job's tears/rice (brightening & balancing), winter melon (acne & anti-aging), rose (skin conditioning), edelweiss/purslane (antioxidants), beta-glucan/aronia berry (wound healing & collagen synthesis), anise (anti-wrinkle), East Asian mountain yam (for fighting collagen loss & reducing skin flakiness), and centella asiatica (anti-inflammatory & anti-bacterial). » Buy Product

Anti-Aging/EGF Serum

Abib Hypoderma SP1-2GF Serum Cell Repair

I initially got this 50% off, but even with a retail value of ~$92 USD, you are definitely paying for quality and advance scientific research with this serum. I still love my Peach & Lily Glass Skin Refining Serum, but this is like a more super-charged anti-aging version. This contains 10 ppm of SP1-EGF (Epidermal Growth Factor) and 15 ppm of SP1-FGF (Fibroblast Growth Factor). The SP1 is a special technology to help skin absorb the EGF/FGF protein molecules. While I don't have too many issues in regards to wrinkles and loss of firmness, after at least 1 month of use, this has made my skin more even (less redness), more baby soft to the touch, more bouncy/resilient, etc. You can say this does slowly reset your skin to a more youthful/baby-ish state. This serum even helped lightened these pesky melasma-like dark spots I had around my cheeks, and have made the minor pores I had on my nose nearly disappear. Other interesting ingredients include bamboo extract (helps absorb nutrients & boost collagen synthesis), niacinamide/honeysuckle (brightening & balancing), Japanese plum (glowing skin & eczema), centella asiatica/propolis (anti-inflammatory & anti-bacterial), and lotus root (heals dry & flaky skin). » Buy Product

Firming/Softening Serum

Peach & Lily Glass Skin Refining Serum

If you are looking for a great multi-tasking serum that will even out, plump, and make your skin smooth (glass skin effect), look no further than the appropriately named Glass Skin Refining Serum. I only use Curology at least once a week and my skin seems to look the best the 2nd day after usage. With the Peach & Lily Glass Skin Refining Serum, I feel like I get that “freshly exfoliated” look every day. This serum contains Peach & Lily's properietary hyaluronic acid complex which according to them a "...blend of short-chain, medium-chain and long-chain hyaluronic acid molecules helps provide hydration at every level of skin; short-chain molecules sink into the deeper layers, while medium-chain molecules fill in the layers closer to the surface of the skin, and long-chain molecules lock in moisture on the skin's surface so that skin is plumped up from within while the outer layer becomes radiant." Other interesting ingredients include niacinamide (brightening & balancing), peach extract (fatty acids & firming), East Asian mountain yam (fighting collagen loss & reducing skin flakiness), madecassoide/beta-glucan (anti-inflammatory & collagen synthesis), and peptides (firming). » Read My Full Review » Buy Product

Anti-Aging/Firming Serum

The Plant Base Time Stop Collagen Ampoule

From my Klog article..."I woke up with really soft skin the next morning. The ampoule absorbs really quickly onto your skin and the softness just felt more noticeable after washing my face.” This is also a product that I’m sure helps with firming and elasticity overtime (one of the few Korean products that contains peptides). It contains 76.52% mushroom extract to stimulate collagen production and reduce the appearance of wrinkles. Other interesting ingredients include mulberry (antioxidant), prickly pear/macadamia nut oil (softening & fatty acids), licorice (skin conditioning & brightening), persimmon/aloe vera/centella asiatica (anti-inflammatory & anti-bacterial), beta-glucan/retinol (healing & collagen synthesis), kakadu plum (vitamin C & acne), and rapeseed (vitamin E). » Read My Full Review » Buy Product

Moisturizer

Abib Hydration Gel Water Tube

One of the most lightweight hydrating gels I’ve ever used. Unless I layer a lot of hydrating products underneath, I never get that wet/oily “just applied moisturizer and I’m waiting for it to absorb” look. This is a moisturizer I definitely would not mind using if I’m just going to be barefaced. Even if I use this at night, I wake up looking almost the same (slightly glowy and hydrated) as before I went to sleep. In other words, I don’t wake up looking all tired and zombie-like. I also don’t oil up too much (if any) with this moisturizer during the day (unlike others) so this is also a good choice for warmer temps/Summer time. Interesting ingredients include red seaweed (hydration & collagen synthesis), sugarcane (antioxidants), meadowfoam (balancing & moisturizing), mushroom/niacinamide (anti-aging & brightening), cactus flower (anti-inflammatory), and moringa (firming & anti-bacterial). » Buy Product

Facial Oil

Peach & Lily Pure Beam Luxe Oil

This is a nice facial oil that doesn't feel super oily. In fact, it feels light (yet still hydrating) enough that if you feel like you need more hydration, you can very easily add more layers. Even with the additional layers, it doesn't go full blown greasy. This oil is 100% natural & plant-based and is more affordable compared to many facial oils that seem as expensive as a high-end serum or cream. Ingredients includes jojoba seed oil (balancing & repairing), squalane (antioxidants & moisturizing), grape seed oil (anti-bacterial & brightening), camellia (antioxidant & omega-6 fatty acids), and sea buckthorn oil (anti-aging & healing). » Buy Product If you have any questions about any of the products above, please let me know. Photo Credits: Soko Glam/Sephora/Peach & Lily

[caithyra]

Commander Jane Shepard 9P3.TCR.R1B.J74.T1U.Q3J.1DS.71E.P37.41I.2GF.9BG.1