Halving the Perimeter of a Zigzag Polygon

A a closed zigzag polygon is made from two isosceles triangles with base 1 and altitude , and a similar triangle with base 2. A straight line through the origin and inclined at an angle cuts the polygon into two parts. What is the length of the blue part?

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DETAILS

Let be the intersection of the lines and . The triangle is congruent to the triangle , and the line cuts the rhombus in half.
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