How Can Vector Proofs Simplify Solving Complex Geometry Problems?

matrix_204
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I was having some serius problems when proving some of the questions where we are given, let's say, a rectangle, there is one diagnol, and the other diagonal is connected to a line that is in a ratio, and the diagnal connects to the point that divides that line. The concept is combined with division of a line segment and geometric proofs, so how can i solve these types of problems, and they take a long time as well.
 
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Sorry, but this makes no sense. The second diagonal is "connected to a line that is in a ratio". How is it "connected"? Do you mean that the second diagonal crosses the line? How is the line "in a ratio"? Do you mean that the line is divided into two segments whose lengths have some specific ratio? By the diagonal?
 

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