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Handwritten math notes show a geometric proof involving a triangle, its incircle, and incenter. The derivation uses formulas for IA, IB, and inradius 'r' to establish the relationship IA · IB = IC · AB, as stated in the article.

AB × CI = AI × BI, part 3

This proof starts with the relations of IA and IB, where I is the incenter of the right triangle, and uses algebraic manipulation to get AB × IC. #math #maths #mathematics #geometry #trigonometry

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A handwritten math proof on lined paper, featuring a right triangle with an inscribed circle and labeled sides/angles. The page shows trigonometric equations and algebraic steps deriving the relationship CI * AB = AI * BI.

AB × CI = AI × BI, part 2

This is another proof of the following right triangle fact that uses some trigonometry. #math #maths #mathematics #geometry #trigonometry

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The 2026 Style Affirmations I’m Living By 🕊✨

Is it just me, or is the constant cycle of "viral trends" getting a little exhausting? 🫠 Lately, I’ve been feeling the need to tune out the noise and get back to what actually makes me feel like me. So, I sat down and wrote these Style Clarity Affirmations for 2026. No more impulse buys ju

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A hand-drawn diagram of an oblique triangle with an inscribed circle and excenters, accompanied by mathematical formulas for triangle area and derivations for half-angle sine and exradii IB and IC, written on lined notebook paper.

Mathematical formulas on lined paper showing the half-angle trigonometric ratios for angle B, including sin(B/2), cos(B/2), tan(B/2), csc(B/2), sec(B/2), and cot(B/2) in terms of the triangle's sides.

Mathematical formulas on lined paper showing the half-angle trigonometric ratios for angle C, including sin(C/2), cos(C/2), tan(C/2), csc(C/2), sec(C/2), and cot(C/2) in terms of the triangle's sides.

Half-angle trig ratios for the other angles

In an oblique triangle, the six trigonometric ratios can be found in terms of the sides. These are for the angles labeled B and C. #math #maths #mathematics #geometry #trigonometry

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A woman in athletic wear stands in a locker room, surrounded by various foods and supplements like steak, salmon, eggs, tofu, protein chips, Greek yogurt, and tart cherry juice, illustrating foods for muscle growth and glute development.

Text highlights protein as the building block for glute gains, listing lean meats, fish, plant-based proteins, and protein powders. Images show steak, a fried egg, tofu cubes, and salmon with lemon slices.

Text emphasizes carbohydrates as fuel for workouts, recommending complex carbs like sweet potatoes, oats, brown rice, and fruits. Partial images include white rice, a loaf of bread, and a bag of protein chips.

Foods for Muscle Growth: Get That Big, Plump Booty

If you’re serious about growing your glutes, you already know that your workouts need to be on point. But did you know that what you eat is just as important as those heavy hip thrusts? Building a bigger, rounder booty isn’t only about spending hours in the gym doing squats and lunges – it’s ab

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Stop procrastinating!

Confession time: I used to be the *queen* of procrastination 👑 Seriously, I'd put off anything that wasn't immediately gratifying. But I knew I had to change, because constantly feeling overwhelmed and behind wasn't the life I wanted. Turning things around wasn't easy, but I disc

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A flat lay of study materials including a notebook with a brain drawing, a pen, a pencil case, an open textbook on a tablet, and a cup of coffee, with the text "Want to remember anything? Try these 6 tricks."

This image illustrates '1. Use Mnemonics' and '2. Chunking' with text explanations and multiple small photos of organized study desks, notebooks, and a coffee cup.

This image illustrates '3. Visualization' and '4. Teach It'. It includes text explanations, photos of study setups, and a 'Learning Pyramid' diagram showing knowledge retention rates.

How to Memorize Anything (Yes, Even Boring Stuff)

1.Make It Fun with Mnemonics 🎶 Mnemonics are like memory magic! Try turning info into catchy acronyms, songs, or even rhymes. Think of it like creating a little “cheat code” for your brain. For example, I could never remember the order of operations in math (who could?!), until I learned “PEMDAS”

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Word problem leading to simple equation

Generated by AI

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A handwritten mathematical proof on lined paper, showing a right-angled triangle ABC with its incircle and inradius 'r'. The proof uses the semiperimeter 's' and algebraic steps to derive the identity IB * IC = (AB - AC) * IA, relating segments from the incenter to the vertices.

IC × IB = (AB - AC) × IA, part 2

This proof starts with the inradius of the right triangle and an identity about a segment connecting the incenter to the medium-sized angle. This proof seems more natural given that the semiperimeter is used more with oblique triangles than right triangles. #math #maths #mathematics #g

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A handwritten mathematical proof on lined paper, showing a right triangle with an inscribed circle and its incenter. The derivation leads to the formula IC • IB = (AB - AC) • IA, demonstrating a relationship between segments connected to the incenter.

IC × IB = (AB - AC) × IA, part 1

In a right triangle, there is an interesting relationship between the segments that connect from the vertices to the incenter (labeled I). The product of the distances of the two shorter segments is equal two the distance of the longest segment multiplied by a factor that itself is the difference b

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A hand holds the book "The Psychology of Money" by Morgan Housel, a Barnes & Noble exclusive edition with a bonus chapter. The cover features a brain made of dollar bills, with a bookshelf and a blurred person in the background.

The Psychology of Money 💰

1 Month of Reading 📖 Summarized in a 5 Minute Summary Title & Author: The Psychology of Money — Morgan Housel * Core Theme: Financial success is driven more by behavior, emotions, patience, and psychology than intelligence or technical knowledge. * Problem It Solves: Why smart people

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A handwritten mathematical proof on lined paper demonstrates that any number of the form 4n+3 cannot be expressed as the sum of two squares. It uses modular arithmetic to analyze the possible remainders of squares (even, odd, or mixed) when divided by 4.

Any number of the form 4n+3 cannot be expressed as the sum of two squares.

This is a fact that comes from elementary number theory. If you have a number that has a remainder of 3 when divided by 4, it can’t be written as a sum of two square numbers. Numbers like 7, 11, 15 etc. can’t be written in that form.

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A plate with a chicken croissant sandwich, watermelon slices, baby carrots, hummus, tortilla chips, and almonds, alongside a glass of water. Overlay text reads "LOSE WEIGHT WHILE EATING Get control over your diet" with "Lemon8" and "@chalie_baker" at the bottom.

Eat Better, Not Less: The 🔑 to Lasting Weight Loss

Weight loss doesn’t mean deprivation or constant hunger—it’s about understanding how to nourish your body while maintaining a calorie deficit. This guide will provide an in-depth look at the roles of macronutrients, portion control, meal timing, calorie management, healthy food swaps, and the s

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A handwritten mathematical derivation on lined paper showing the distance between the orthocenter (O) and circumcenter (P) of a triangle. It uses the property a²+b²+c²=7R² to simplify Euler's theorem, concluding that OP = R√2 for a heptagonal triangle.

Simplifying the distance between two points in the heptagonal triangle

In a prior post, I derived the distance formula between the orthocenter and the circumcenter using a complicated argument involving similar triangles. In this post, I used a simple formula to get the required answer. #math #maths #maths #geometry #trigonometry

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A woman in lavender workout attire stands in a gym, holding her long dark hair. Text overlay reads 'equation for the perfect GLUTE DAY', introducing a glute workout routine.

A woman performs a hip thrust exercise on a specialized machine in a gym, wearing lavender workout clothes and headphones. Text overlay indicates this movement as 'THRUST'.

A woman performs a barbell squat in a squat rack, facing away from the camera, showcasing her form. Text overlay identifies this exercise as 'SQUAT' for a glute workout.

equation for the perfect glute day💜

there’s no need to overcomplicate things! when it comes to an effective glute day, i think i have perfected the equation with just a few movements!! there are multiple ways to perform these movements so let’s go over some! thrust + squat + hinge + isolation = a juicy pump thrust: - hip t

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A person in workout attire stands next to text stating "CARBS DON'T MAKE YOU FAT!". Surrounding them are images of Fairlife chocolate milk, pasta with broccoli, bread, white rice, avocado, and ice cream, illustrating various food types.

A graphic titled "What Should I Eat? Foods with High Quality Carbs" displays images of butternut squash, granola, couscous, sweet potatoes, pasta, lentils, oats, cooked rice, and fruit, each with their carbohydrate content per serving.

Text outlines how to calculate carbohydrate needs for weight loss, including formulas for Basal Metabolic Rate (BMR) for men and women. A yellow cartoon character is shown thinking of a croissant.

Carbs are not the enemy!!✋🏼✨🩵 Weight loss With Carb

I wanted to talk about something that's been on my mind lately—carbohydrates. It seems like carbs have gotten a bad reputation, especially when it comes to weight loss. There's this belief out there that if you want to shed pounds, you have to avoid carbs at all costs. But let me tell y

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A handwritten math solution on lined paper, featuring a triangle diagram with orthocenter (O) and circumcenter (P) labeled. It derives the formula for the squared distance between them, d² = 9R² - (a²+b²+c²), using trigonometric identities and cosine rules.

The distance between the orthocenter and the circumcenter of a triangle

In any acute triangle, the distance between the orthocenter and the circumcenter is found with quite a nice formula that relate the sides and the circumradius. #math #maths #mathematics #geometry #trigonometry

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A handwritten page showing mathematical formulas for triangle area and sine ratios. It includes Heron's formula, area formulas using sine, and derivations for sin A, sin B, sin C, and csc A, csc B, csc C in terms of the triangle's sides and semi-perimeter.

Sine ratios in terms of the triangle’s sides

In any oblique triangle, the sine ratios can be found in terms of the sides using the given formulae. #math #maths #mathematics #geometry #trigonometry

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A handwritten mathematical derivation of the trigonometric identity cos(A+B) = cos A cos B - sin A sin B. The derivation uses a triangle diagram, area formulas, and various trigonometric substitutions, progressing step-by-step to the final identity.

cosA × cos B - sin A × sin B = cos(A+B)

This is an obscure derivation of a well-known trigonometric identity. #math #maths #mathematics #geometry #trigonometry

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Two halves of a red grapefruit are displayed on a wooden cutting board. Overlaying the image is text that reads "LOSE WEIGHT WHILE EATING How to structure your diet*" in yellow and white fonts.

A white background with black text titled "UNDERSTANDING YOUR MACROS." The text explains that macronutrients (carbohydrates, proteins, fats) are the building blocks of diet, each serving a unique purpose for energy, muscle, and hormone production, essential for weight loss.

A white background with black text titled "HOW TO DETERMINE YOUR MACROS." It outlines steps for calculating macros: determining caloric needs, setting macro ratios (e.g., 40% carbs, 35% protein, 25% fats for weight loss), and adjusting based on results.

How to Eat to Lose Lbs: Diet for Maximum Fat Burn🔥

Embarking on a weight loss journey can be overwhelming, especially when it comes to diet. There’s so much conflicting information out there—low carb, high protein, intermittent fasting, clean eating—the list goes on. But the truth is, successful weight loss isn’t about depriving yourself or jumping

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A woman in pink athletic wear and headphones poses in a gym, showcasing her hamstrings. Text overlay reads "3 EXERCISES FOR HAMSTRING GROWTH" and "Peach Gym". An arrow points to her hamstring.

A woman in pink athletic wear performs sumo squats with a barbell in a gym. Text overlay reads "1. sumo squats". The gym features pink walls and black equipment.

A woman in pink athletic wear performs hip thrusts with a barbell, using a bench in a gym. Text overlay reads "2. hip thrusts" and "Big Peach Gym".

the hamstring equation i swear by💪🏽

i LOVE a good hamstring pump! i’ve been working hard the past few years to grow them and i finally see it paying off! a few of my current favorites include: - sumo squats (landmine, smith machine, dumbell) - hip thrusts (landmine, smith machine, barbell, hip thrust machine) - single leg h

47 likes

A handwritten mathematical derivation on lined paper, showing the steps to find Pythagorean triples. It starts with a right-angled triangle labeled with sides a, b, and hypotenuse c, then derives the formulas a = m² - n², b = 2mn, and c = m² + n² from the Pythagorean theorem, with an example calculation for m=2, n=1 resulting in 3, 4, 5.

Deriving the formula to find Pythagorean triples

Given natural number m, n such that m > n > 0, then the derived formulae can be used to find Pythagorean triples. Pythagorean triples are sets of three whole numbers that satisfy the Pythagorean theorem. #math #maths #mathematics #geometry #trigonometry

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Easy Habits to Boost Weight Loss

Losing weight can feel really overwhelming, especially with a million voices on social media telling you that this skinny tea is the secret or that morning routine holds the secret to finally losing weight. But the secret is that there is no secret. Weight loss is a simple equation…. Calories in

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The image illustrates solving quadratic equations by factoring. It shows the general form X²+bx+C=0, its factored form (X-r)(X-s)=0, and an example X²-5x+6=0. The example shows (X+3)(X+2)=0 leading to X=-3 or X=-2, alongside an X-factor diagram and verification calculations.

solving quadratic equation using factor 🫶

#math #mathnotes #lemon8edu

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Proving the existence of the Gergonne point of a triangle

The Gergonne point is where three specific segments are concurrent. These segments come off the vertices and intersect where the incircle is tangent to the side opposite of the vertex. While not as well-known as the most well-known triangle centers, its existence is easily proven with Ceva’s theore

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WHEN THE 100K ARM FAILS BUT THE 10 DOLLAR STRAP WINS This is the unfiltered reality of training with a prosthetic arm. The first clip shows exactly what happens when the locking mechanism gives out during a heavy RDL. Gravity takes over and the set is ruined. While my bionic arm is a masterpiec

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A woman in athletic wear takes a mirror selfie in a gym, with text overlay introducing "3 HIGH PROTEIN MEAL IDEAS" that are simple, 30-minute, affordable, and good for meal prep to grow muscles and fuel balanced eating.

A close-up of a ground turkey and pasta dish topped with cottage cheese and pepper, alongside the recipe's ingredients and directions, indicating approximately 40g of protein per serving.

A close-up of a taco pasta salad meal prep dish with ground turkey, pasta, and vegetables, accompanied by its ingredients and directions, noting approximately 36g of protein.

3 HIGH Protein Recipes to FUEL Good 💪🏼💗

High protein meals need to be a priority if you want to gain muscle and lose fat… period. If your body isn’t getting enough protein it won’t be able to do its job and aid in muscle recovery, which is basically what helps to grow the muscles that you see. ➡️ A good rule of thumb is to consume .5

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A handwritten derivation of the cosine half-angle identity on lined paper. It features a right-angled triangle with an angle bisector, labeled sides and angles, and algebraic steps leading to the formula cos(θ/2) = sqrt( (1/2)(cos θ + 1) ).

Proving the cosine half-angle identity

This is proven using the angl bisector theorem, Pythagorean theorem and algebra. #math #maths #mathematics #geometry #trigonometry

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Sides of a triangle in cosine form

Every acute triangle can have its sides written in terms of the other sides and the cosines of the angles adjacent to that side being solved for. #math #maths #mathematics #geometry #trigonometry

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WHEN THE 100K ARM FAILS BUT THE 10 DOLLAR STRAP WINS This is the unfiltered reality of training with a prosthetic arm. The first clip shows exactly what happens when the locking mechanism gives out during a heavy RDL. Gravity takes over and the set is ruined. While my bionic arm is a masterpiec

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A desk with a laptop, notebook, and phone displaying "How to study for your type of brain." The main title reads "How to Memorize Anything: Tailored Techniques for Every Type of Brain," accompanied by a pink brain icon.

This section details "1. Spaced Repetition," explaining it as reviewing information at increasing intervals. A diagram illustrates the process of receiving, recalling, and retaining content through four steps. It's best for short-term and semantic memory brains.

This section explains "2. Mnemonic Devices," memory aids using associations. Examples at the top show visualization (fish skeleton), storytelling (rabbit), and acronyms (HOMES for Great Lakes). It's best for semantic and episodic memory brains.

How to Memorize ✨ANYTHING✨: Tips for ALL 🧠 Types!

✨First Things First: Get to know your brain 🧠!✨ 🧠1. ADHD Brain: The ADHD brain struggles with sustained focus but thrives on novelty and excitement. It processes information quickly but often gets distracted or hyper-focused on areas of interest. 🧠2. Sensory Memory Brain: This brain type is h

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Handwritten mathematical derivations on lined paper, showing the step-by-step process to derive the triangle area formula Δ = 2R² sin A sin B sin C from Δ = abc / 4R, using the sine rule substitutions for a, b, and c.

Δ = 2R² × sin A × sin B × sin C

These are area formulae for any triangle given the side lengths, the circumradius and the angle measurements. #math #maths #mathematics #geometry #trigonometry

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An infographic titled 'The Accounting Equation' explaining Assets = Liabilities + Owners Equity. It details each component with definitions, debit/credit rules, impact of income/expenses, cash flow formulas, importance, what high/low numbers represent, general targets, and examples like cash, inventory, credit cards, loans, common stock, and retained earnings.

Day 1/60 Road to CFO: The Accounting Equation

Accounting Fundamentals: The Accounting Equation Often times the first thing you’ll learn in a college Accounting course… and many would say it’s the most important concept in Accounting (hence the name) But what’s so special about it? Let’s dive in. ➡️ What the idea? This equati

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A notebook page displays a geometric diagram of a right triangle with its incenter and inscribed circle, alongside a detailed algebraic proof. The calculations demonstrate that the product of the hypotenuse and the bisector from the right angle equals the product of the bisectors from the acute angles.

AB × CI = AI × BI, part 1

In a right triangle with the incenter drawn, the bisectors coming off the smaller angles are longer than the one coming off the right angle. The product of the distances of the longer bisectors is equal to the product of the length of the short bisector and the length of the hypotenuse. #mTh

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The Edge Of Knowing

The Edge Of Knowing Maybe humanity was never meant to have all the answers. Maybe the beauty was always in the search itself. We were born from a universe vast enough to create stars, oceans, gravity, mathematics, consciousness, and memory… and somehow, from all of that, something learne

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Proving the Law of Cosines

This post shows a standard proof of the Law of Cosines in any oblique triangle. #math #maths #mathematics #geometry #trigonometry

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The equation of relationships

Once you figure out the science behind relationships, it makes everything much more simple

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A bright yellow building under a blue sky, framed by green trees, with the text "STEM 101" overlaid, introducing the topic of STEM education.

A laptop screen displaying a whimsical cat and fish pond scene, with the text "STEM = Science, Tech, Engineering, and Math" overlaid, defining the acronym.

A petri dish with bacterial growth, labeled for biology, accompanied by an overlay listing "Science: - Biology - Chemistry - Physics etc." as STEM fields.

STEM Made Simple: What It Really Means

A big thank you to the lovely person who asked me about STEM—it gave me the perfect excuse to share this little explainer post to you all. 💕 When I first heard the word “STEM,” it felt so big and intimidating… like this giant world of smart people doing impossible things. But the truth is, STEM

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🧮The Jump Higher Equation 🍋

1. Pick a plyometric activity box jumps vertical jumps you pick. 2. Find a weight you can move for just five reps and move it fast. 3. Combine them together to make a contrast set and perform a twice a week on your low body day. Not only will you jump higher, but your butt will look better too

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A handwritten mathematical proof of the sine half-angle identity, sin(θ/2) = √[1/2(1-cosθ)], shown on a lined notebook page. The proof uses a geometric diagram of a right-angled triangle with an angle bisector and algebraic derivations.

Proving the sine half-angle identity

This identity is proven, using the angle bisector theorem, the Pythagorean theorem and algebra. #math #maths #mathematics #geometry #trigonometry

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A handwritten note explaining mixture problems in differential equations, defining rate in/out, and solving an example where a 1000L tank with 100kg sugar is diluted with pure water until only 10kg sugar remains, calculating the time as 460 seconds.

Mixture problem in Differentiable equation

#mathematics #lemon8education #notes

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Put God in your daily equation.When God is added to the equation, confusion find

Put God in your daily equation. When God is added to the equation, confusion finds clarity, weakness finds strength, and obstacles become opportunities. Don’t try to solve life’s problems with your own understanding alone. Start every day with prayer, trust His direction, and watch how He multip

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A book cover titled "DON'T WORRY" by Shunmyo Masuno, featuring "48 Lessons on Relieving Anxiety" from a Zen Buddhist monk, with pink and green circles.

A pink and purple boba tea-shaped pocket journal with a winking face, brown dots, a striped straw, and a small pink heart lock, accompanied by pink fluffy pens.

5 below finds Journaling Edition ✨🌸 Part 1

💵 Price: $5.55 Purchased from: 5 below How to be the main Character - I thought this would be a cute little journal, to remind myself that I matter as well, sometimes we get so caught up with life it’s okay to pause and remember you’re an important part of the equation. 💛❤️ Kawaii Black Ink

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A handwritten mathematical derivation on lined paper, showing a right triangle with an inscribed circle. It illustrates the relationship between the triangle's sides (a, b, c) and inradius (r), deriving the identity r(c+r) = (a-r)(b-r) through area calculations.

r(c+r) = (a-r)(b-r)

Given a right triangle, the following identities are true. #math #maths #mathematics #geometry #trigonometry

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A woman with dark hair, wearing a striped turtleneck sweater and jeans, sits looking at the camera. The image has text overlays that read "wellness hacks" and "lemon8 @isa.qadira", serving as a cover for wellness tips.

A serene beach scene features ocean waves crashing on the shore next to a rocky cliff under a clear blue sky. An overlay text box lists wellness tips for low energy, bloating, stress, and cramps.

An indoor view shows multiple disco balls hanging from a wooden ceiling, with some green plants visible. An overlay text box presents wellness tips for soreness, brain fog, difficulty sleeping, and sadness.

the wellness equation

here’s some quick tips for health and wellness! of course it’s not always this simple but i would say it’s highly likely that these actionable habits will make you feel better. what has worked for you all? i’d love to hear about what you think i’m missing or if anyone has some tips that works reall

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bc³ - a³c - ab³ = 7R⁴

This is a proof of a relation in the heptagonal triangle. Each variable represents the lengths of the sides such that a < b < c. R is the length of the circumradius. #math #maths #mathematics #geometry #trigonometry

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A handwritten diagram of an oblique triangle ABC with sides a, b, c and altitudes ha, hb, hc. The image shows the derivation of the sine area formulae for a triangle, including Δ = ½ ac sin B, Δ = ½ ab sin C, and Δ = ½ bc sin A, proving the area calculation using trigonometry.

Proving the sine area formulae

This post proves the sine area formulae for any oblique triangle. #math #maths #mathematics #geometry #trigonometry

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A cozy flat lay featuring the book cover for "Rekindling THE Flame" by Madison Myers, an open book, a coffee mug, cinnamon rolls, and autumn leaves on a gray knitted sweater, evoking a fall reading atmosphere.

Rekindling the flame 💚🌲🍀

I never intended to set foot in Smoky Heights again. But when I learn my mom has less than a year to live, I swallow my pride and return to help her through her final months. My plan is simple: stay focused, avoid the past, and most importantly, steer clear of Wyatt Grady—my first love and the o

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