document: Vector algebra proofs for geometry theorems Vector mathematics can generate simple and powerful proofs of theorems in plane geometry . These proofs can also be used to generalize plane geometry theorems to higher dimensions . We present three vector proofs that show the power of this technique . 1 . For any quadrilateral , the sum of the squares of the diagonals is less than or equal to the sum of the squares of the sides . 2 . The area of a quadrilateral is half the product of the diagonals multiplied by the sine of an included angle . 3 . One quarter of all triangles are acute -LRB- Based upon the options detailed below , with respect to the relative lengths of the sides -RRB- . This paper presents a set of examples of vector mathematics applied to geometry problems . Some of the most beautiful and sophisticated proofs in mathematics involve using multiple representations of the same data . By leveraging the advantages of each representation one finds new and useful mathematical facts
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extractive_keyphrases: vector mathematics plane geometry multiple representations quadrilateral proofs
abstractive_keyphrases: vector algebra proofs