# Difference between revisions of "Wooga Looga Theorem"

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+ | ==Application 1== | ||

+ | ===Problem 1=== | ||

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

In <math>\triangle ABC</math> points <math>X,Y,Z</math> are on sides <math>BC,CA,AB</math> such that <math>\frac{XB}{XC}=\frac{YC}{YA}=\frac{ZA}{ZB}=\frac 71</math>. Find the ratio of <math>[XYZ]</math> to <math>[ABC]</math>. | In <math>\triangle ABC</math> points <math>X,Y,Z</math> are on sides <math>BC,CA,AB</math> such that <math>\frac{XB}{XC}=\frac{YC}{YA}=\frac{ZA}{ZB}=\frac 71</math>. Find the ratio of <math>[XYZ]</math> to <math>[ABC]</math>. | ||

− | + | ===Solution 1=== | |

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> 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>. 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> 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>. Therefore the answer is <math>\boxed{\frac{43}{64}}</math>. | ||

− | + | ==Application 2== | |

− | + | ===Problem 2=== | |

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

The figure below shows a triangle ABC which area is 72cm2. If AD: DB = BE: EC =CF: FA =1: 5, find the area of triangle DEF | The figure below shows a triangle ABC which area is 72cm2. If AD: DB = BE: EC =CF: FA =1: 5, find the area of triangle DEF | ||

− | + | ===Solution 2=== | |

is this solution by franzliszt: | is this solution by franzliszt: | ||

## Revision as of 11:51, 29 October 2020

## Application 1

### Problem 1

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 and , , . Then . Therefore the answer is .

## Application 2

### Problem 2

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

The figure below shows a triangle ABC which area is 72cm2. If AD: DB = BE: EC =CF: FA =1: 5, find the area of triangle DEF

### Solution 2

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 .

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