# Difference between revisions of "Wooga Looga Theorem"

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Proof by franzliszt | Proof by franzliszt | ||

− | Apply Barycentrics w.r.t. <math>\triangle ABC</math>. Then <math>A=(1,0,0),B=(0,1,0),C=(0,0,1)</math>. We can also find that <math>D=(0,\tfrac {1}{r+1},\tfrac {r}{r+1}),E=(\tfrac {r}{r+1},0,\tfrac {1}{r+1}),F=(\tfrac {1}{r+1},\tfrac {r}{r+1},0)</math>. In the barycentric coordinate system, the area formula is <math>[XYZ]=\begin{vmatrix} x_{1} &y_{1} &z_{1} \\ x_{2} &y_{2} &z_{2} \\ x_{3}& y_{3} & z_{3} \end{vmatrix}\cdot [ABC]</math> where <math>\triangle XYZ</math> is a random triangle and <math>\triangle ABC</math> is the reference triangle. Using this, we find that <cmath>\frac{[DEF]}{[ABC]}= \begin{vmatrix} 0&\tfrac {1}{r+1}&\tfrac {r}{r+1} \\ \tfrac {r}{r+1}&0&\tfrac {1}{r+1}\\ \tfrac {1}{r+1}&\tfrac {r}{r+1}&0 \end{vmatrix}=\frac{r^2-r+1}{(r+1)^2}.</cmath> | + | Apply Barycentrics w.r.t. <math>\triangle ABC</math>. Then <math>A=(1,0,0),B=(0,1,0),C=(0,0,1)</math>. We can also find that <math>D=\left(0,\tfrac {1}{r+1},\tfrac {r}{r+1}\right),E=\left(\tfrac {r}{r+1},0,\tfrac {1}{r+1}\right),F=\left(\tfrac {1}{r+1},\tfrac {r}{r+1},0\right)</math>. In the barycentric coordinate system, the area formula is <math>[XYZ]=\begin{vmatrix} x_{1} &y_{1} &z_{1} \\ x_{2} &y_{2} &z_{2} \\ x_{3}& y_{3} & z_{3} \end{vmatrix}\cdot [ABC]</math> where <math>\triangle XYZ</math> is a random triangle and <math>\triangle ABC</math> is the reference triangle. Using this, we find that <cmath>\frac{[DEF]}{[ABC]}= \begin{vmatrix} 0&\tfrac {1}{r+1}&\tfrac {r}{r+1} \\ \tfrac {r}{r+1}&0&\tfrac {1}{r+1}\\ \tfrac {1}{r+1}&\tfrac {r}{r+1}&0 \end{vmatrix}=\frac{r^2-r+1}{(r+1)^2}.</cmath> |

==Proof 3== | ==Proof 3== | ||

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is this solution by RedFireTruck: | is this solution by RedFireTruck: | ||

− | WLOG let <math>A=(0, 0)</math>, <math>B=(1, 0)</math>, <math>C=(x, y)</math>. Then <math>[ABC]=\frac12|y|</math> by Shoelace Theorem and <math>X=(\frac{7x+1}{8}, \frac{7y}{8})</math>, <math>Y=(\frac{x}{8}, \frac{y}{8})</math>, <math>Z=(\frac78, 0)</math>. Then <math>[XYZ]=\frac12|\frac{43y}{64}|</math> by Shoelace Theorem. Therefore the answer is <math>\boxed{\frac{43}{64}}</math>. | + | WLOG let <math>A=(0, 0)</math>, <math>B=(1, 0)</math>, <math>C=(x, y)</math>. Then <math>[ABC]=\frac12|y|</math> by Shoelace Theorem and <math>X=\left(\frac{7x+1}{8}, \frac{7y}{8}\right)</math>, <math>Y=\left(\frac{x}{8}, \frac{y}{8}\right)</math>, <math>Z=\left(\frac78, 0\right)</math>. Then <math>[XYZ]=\frac12\left|\frac{43y}{64}\right|</math> by Shoelace Theorem. Therefore the answer is <math>\boxed{\frac{43}{64}}</math>. |

==Solution 2== | ==Solution 2== | ||

or this solution by franzliszt: | or this solution by franzliszt: | ||

− | We apply Barycentric Coordinates w.r.t. <math>\triangle ABC</math>. Let <math>A=(1,0,0),B=(0,1,0),C=(0,0,1)</math>. Then we find that <math>X=(0,\tfrac 18,\tfrac 78),Y=(\tfrac 78,0,\tfrac 18),Z=(\tfrac18,\tfrac78,0)</math>. In the barycentric coordinate system, the area formula is <math>[XYZ]=\begin{vmatrix} | + | We apply Barycentric Coordinates w.r.t. <math>\triangle ABC</math>. Let <math>A=(1,0,0),B=(0,1,0),C=(0,0,1)</math>. Then we find that <math>X=\left(0,\tfrac 18,\tfrac 78\right),Y=\left(\tfrac 78,0,\tfrac 18\right),Z=\left(\tfrac18,\tfrac78,0\right)</math>. In the barycentric coordinate system, the area formula is <math>[XYZ]=\begin{vmatrix} |

x_{1} &y_{1} &z_{1} \\ | x_{1} &y_{1} &z_{1} \\ | ||

x_{2} &y_{2} &z_{2} \\ | x_{2} &y_{2} &z_{2} \\ |

## Revision as of 00:27, 6 November 2020

## Contents

# Definition

If there is and points on the sides respectively such that , then the ratio .

Created by the Ooga Booga Tribe of the Caveman Society, https://www.youtube.com/channel/UC50E9TuLIMWbOPUX45xZPaQ

# Proofs

## Proof 1

Proof by Gogobao:

We have:

We have:

Therefore

So we have

## Proof 2

Proof by franzliszt

Apply Barycentrics w.r.t. . Then . We can also find that . In the barycentric coordinate system, the area formula is where is a random triangle and is the reference triangle. Using this, we find that

## Proof 3

Proof by RedFireTruck:

WLOG we let , , for , . We then use Shoelace Forumla to get . We then figure out that , , and so we know that by Shoelace Formula . We know that for all so .

# Application 1

## Problem

The Wooga Looga Theorem states that the solution to this problem by franzliszt:

In points are on sides such that . Find the ratio of to .

## Solution 1

is this solution by RedFireTruck:

WLOG let , , . Then by Shoelace Theorem and , , . Then by Shoelace Theorem. Therefore the answer is .

## Solution 2

or this solution by franzliszt:

We apply Barycentric Coordinates w.r.t. . Let . Then we find that . In the barycentric coordinate system, the area formula is where is a random triangle and is the reference triangle. Using this, we find that

## Solution 3

or this solution by aaja3427:

According the the Wooga Looga Theorem, It is . This is

## Solution 4

or this solution by ilovepizza2020:

We use the to instantly get . (Note: You can only use this method when you are not in a contest as this method is so overpowered that the people behind tests decided to ban it.)

## Solution 5

or this solution by eduD_looC:

This is a perfect application of the Adihaya Jayasharmaramankumarguptareddybavarajugopal's Lemma, which results in the answer being . A very beautiful application, which leaves graders and readers speechless.

## Solution 6

or this solution by CoolJupiter:

Wow. All of your solutions are slow, compared to my sol:

By math, we have .

~CoolJupiter

# Application 2

## Problem

The Wooga Looga Theorem states that the solution to this problem by Matholic:

The figure below shows a triangle ABC whose area is . If AD: DB = BE: EC =CF: FA =1: 5, find the area of triangle DEF

## Solution 1

is this solution by franzliszt:

We apply Barycentric Coordinates w.r.t. . Let . Then we find that . In the barycentric coordinate system, the area formula is where is a random triangle and is the reference triangle. Using this, we find that so .

## Solution 2

or this solution by RedFireTruck:

By the Wooga Looga Theorem, . We are given that so

# Application 3

## Problem

The Wooga Looga Theorem states that the solution to this problem by RedFireTruck:

Find the ratio if and in the diagram below.

## Solution 1

is this solution by franzliszt:

By the Wooga Looga Theorem, . Notice that is the medial triangle of **Wooga Looga Triangle ** of . So and by Chain Rule ideas.

## Solution 2

or this solution by franzliszt:

Apply Barycentrics w.r.t. so that . Then . And .

In the barycentric coordinate system, the area formula is where is a random triangle and is the reference triangle. Using this, we find that

# Application 4

## Problem

Let be a triangle and be points on sides and respectively. We have that and similar for the other sides. If the area of triangle is , then what is the area of triangle ? (By ilovepizza2020)

## Solution 1

By Franzliszt

By Wooga Looga, so the answer is .

## Solution 2

By franzliszt

Apply Barycentrics w.r.t. . Then . We can also find that . In the barycentric coordinate system, the area formula is where is a random triangle and is the reference triangle. Using this, we find thatSo the answer is .

# Testimonials

Franzlist is wooga looga howsopro - volkie boy

The Wooga Looga Theorem is EPIC POGGERS WHOLESOME 100 KEANU CHUNGUS AMAZING SKILL THEOREM!!!!!1!!!111111 -centslordm

The Wooga Looga Theorem can be used to prove many problems and should be a part of any geometry textbook. ~ilp2020

The Wooga Looga Theorem is amazing and can be applied to so many problems and should be taught in every school. - RedFireTruck

The Wooga Looga Theorem is the best. -aaja3427

The Wooga Looga Theorem is needed for everything and it is great-hi..

The Wooga Looga Theorem was made by the author of the 3rd Testimonial, RedFireTruck, which means they are the ooga booga tribe... proof: go to https://www.youtube.com/channel/UC50E9TuLIMWbOPUX45xZPaQ and click "about". now copy and paste the aops URL. you got RedFireTruck! Great Job! now go check out his thread for post milestones, https://artofproblemsolving.com/community/c3h2319596, and give him a friend request! -FPT

This theorem has helped me with school and I am no longer failing my math class. -mchang

"I can't believe AoPS books don't have this amazing theorem. If you need help with math, you can depend on caveman." ~CoolJupiter

Before the Wooga Looga Theorem, I had NO IDEA how to solve any hard geo. But, now that I've learned it, I can solve hard geo in 7 seconds ~ ilp2020 (2nd testimonial by me)

Too powerful... ~franzliszt

The Wooga Looga Theorem is so pro ~ ac142931