Tato Red Envelope for Chinese New Year

By symmetry, the areas of $\bigtriangleup XHG$ and $\bigtriangleup GZF$ are each repeated 4 times in the diagram below. So the basic idea will be to find the area of each triangle, add them, and multiply the result by 4.

Suppose the unit square is oriented on the coordinate plane so $G$ is at (0,0) and $C$ is at (1,1). Given this setup, we first want the coordinates of points $X,$ $Y,$ and $Z.$

The location of $X$ occurs at the intersection of lines $y = 2x$ and $y = -\frac{1}{2}x + \frac{1}{2} .$ Substituting, $2x = -\frac{1}{2}x + \frac{1}{2}$ so $\frac{5}{2}x = \frac{1}{2}$ and therefore $x = \frac{1}{5} .$ This also implies $y = 2x = 2 \cdot \frac{1}{5} = \frac{2}{5}.$ Therefore $X$ is at $\left( \frac{1}{5}, \frac{2}{5} \right) .$

$Y$ is symmetrical to $X$ across the line $y = x$ , so the point is simply the reflection such that the $x$ and $y$ coordinates swap: $\left(\frac{2}{5}, \frac{1}{5}\right) .$

The location of $Z$ occurs at the intersection of lines $y = -x + 1$ and $y = \frac{1}{2}x .$ Substituting, $-x +1 = \frac{1}{2}x$ and so $\frac{3}{2}x = 1$ implying $x = \frac{2}{3} .$ Since $y = -x + 1$ this means $y = \frac{1}{3} .$ Therefore $Z$ is at $\left( \frac{2}{3}, \frac{1}{3} \right) .$

Now we can find the areas using the triangle formula $\frac{1}{2} \cdot \text{base} \cdot \text{height} :$

For the area of $\bigtriangleup XHG ,$ using the base at $GH$ (which has a length of $\frac{1}{2} )$ and height dropping a perpendicular from $X$ to $GH$ (which by the coordinate $X$ has a length of $\frac{1}{5} ),$ the area is $\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{5} = \frac{1}{20} .$

For the area of $\bigtriangleup GZF$ we can use the base at $GZ$ and height at $YF .$ The length of each can be found with the distance formula and the coordinates calculated above. The length of $GZ$ between (0,0) and $\left( \frac{2}{3}, \frac{1}{3} \right)$ is $\frac{\sqrt{5}}{3} .$ The length of $YF$ between $\left(\frac{2}{5}, \frac{1}{5}\right)$ and $\left(\frac{1}{2}, 0\right)$ is $\frac{1}{2\sqrt{5}} .$ Using these to find the area results in $\frac{1}{2} \cdot \frac{\sqrt{5}}{3} \cdot \frac{1}{2\sqrt{5}} = \frac{1}{12} .$

Adding these areas gets $\frac{1}{20} + \frac{1}{20} = \frac{2}{15} ;$ multiplying by 4 (since each triangle occurs 4 times in the patterned area) gets a final result of $\frac{2}{15} \cdot 4 = \frac{8}{15} .$ Therefore the answer desired is $8 + 15 = 23 .$

Use coordinate geometry.

Let G(0,0). The patterns are repeated 4x, so we need only to calculate the area of 1 pattern and then multiply by 4.Look at triangles GFZ, and FEW, where W is the top vertex of the triangle FEW. Coordinates of Z are found by the intersection of lines GD and FC, which are y=1/2 x , and y=2x -1 respectively. So Z(2/3,1/3) . Coordinates of W are found by the intersection of lines EH and FC, which are y=-1/2 x+1/2 , and y=2x- 1 respectively. So Z(3/5,1/5) . So the area = 4 [(1/2)(1/6) + (1/2)(1/10)] = 8/15. So m+n=23

First proof of ratios required is given.
Area is symmetrical , so one quarter area is considered. Quater area is divided into two parts. Each part solved separately and finally added.