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\textbf{Menelaus' Theorem}  A necessary and sufficient condition for points $P, Q, R$ on the respective sides $BC, CA, AB$ (or their extensions) of a triangle $ABC$ to be collinear is that  $BP\cdot CQ\cdot AR = PC\cdot QA\cdot RB$\\  \textbf{Ptolemy's Theorem}  Given a cyclic quadrilateral $ABCD$ with side lengths ${a},{b},{c},{d}$ and diagonals ${e},{f}$:\[ac+bd=ef.\] \\  \textbf{Mass Point Geometry}  Mass points is a technique in Euclidean geometry that can greatly simplify the proofs of many theorems concerning polygons, and is helpful in solving complex geometry problems involving lengths. In essence, it involves using a local coordinate system to identify points by the ratios into which they divide line segments. Mass points are generalized by barycentric coordinates.  Mass point geometry involves systematically assigning 'weights' to points using ratios of lengths relating vertices, which can then be used to deduce other lengths, using the fact that the lengths must be inversely proportional to their weight (just like a balanced lever). Additionally, the point dividing the line has a mass equal to the sum of the weights on either end of the line (like the fulcrum of a lever). \\  \section{Trigonometry}  Basic Definitions $\sin A = \frac ac$ $\csc A = \frac ca$ $\cos A = \frac bc$ $\sec A = \frac cb$ $\tan A = \frac ab$ $\cot A = \frac ba$  Even-Odd Identities