Partitioning a Star

Draw a line as shown in the figure in the inner Pentagon,

Denote the area of the red region within the inner pentagon as b

this is also the area of the pink region within the inner pentagon.

Let the 5 isosceles triangles on the Pentagon have an area of a

Then the blue triangle within the inner Pentagon also has an area of a

The area of the blue region is : 3a + b

The area of the pink region is also 3a + b

Therefore the ratio of the pink region to the blue region is 1.

O.K the ratio is 1:1

How?

1). Since there are no info says otherwise, then this is a regular pentagram which means:

• And let's first divide them into outer triangles and inner ones which are part of the core pentagon

• These outer triangles are isosceles with a head angle (the most outer angle) measure of 36ْ and 2 identical base angles of 72ْ

• The core pentagon has 5 identical angles of 108ْ .. that means the little pink triangle is an isosceles one of head angle 108 and 2 identical base triangles of 36ْ

• Now if merged the small pink triangle with one of the bigger two that shares edge with you can form an even bigger triangle which happens to be an isosceles as well .. how do we know that? well by combining both 1.1 and 1.2 we'll find out that the two base angles that were merged together to form a vertex are both forming a head of measure 108 and the 2 other angles equal 36 each .. which will lead to conclusion that small pink triangle edge that cuts through the pentagon is actually equals to the 2 edges of the outer triangles

2). Now if you draw another line inside the pentagon to form another small triangle that's exactly like the pink one .. you'll find yourself ended up with the core pentagon divided into 3 triangles 2 identical 1 pink and 1 blue and a 3rd bigger one that have the 2 bases of the other 2 triangles as its edges which equals to the edges of the outer triangles

And that means you'll have 3 big blue triangles and 1 small one, exactly like the pinks .. which means they have the same ratio "1"

Both are equal, So the ratio = 1.