Euler's quadrilateral theorem
Euler's quadrilateral theorem or Euler's law on quadrilaterals, named after Leonhard Euler (1707–1783), describes a relation between the sides of a convex quadrilateral and its diagonals. It is a generalisation of the parallelogram law which in turn can be seen as generalisation of the Pythagorean theorem. Because of the latter the restatement of the Pythagorean theorem in terms of quadrilaterals is occasionally called the Euler–Pythagoras theorem. Theorem and special cases. For a convex quadrilateral with sides formula_1, diagonals formula_2 and formula_3, and formula_4 being the line segment connecting the midpoints of the two  diagonals, the following equations holds: If the quadrilateral is a parallelogram, then the midpoints of the diagonals coincide so that the connecting line segment formula_4 has length 0. In addition the parallel sides are of equal length, hence Euler's theorem reduces to which is the parallelogram law. If the quadrilateral is rectangle, then equation simplifies further since now the two diagonals are of equal length as well: Dividing by 2 yields the Euler–Pythagoras theorem: In other words, in the case of a rectangle the relation of the quadrilateral's sides and its diagonals is described by the Pythagorean  theorem. Alternative formulation and extensions. Euler originally derived the theorem above as corollary from slightly different theorem that requires the introduction of an additional point, but provides more structural insight. For a given convex quadrilateral formula_10 Euler introduced an additional point formula_11 such that formula_12 forms a parallelogram and then the following equality holds: The distance formula_14 between the additional point formula_11 and the point formula_16 of the quadrilateral not being part of the parallelogram can be thought of measuring how much the quadrilateral deviates from a parallelogram and formula_17 is correction term that needs to be added to the  original equation of the parallelogram law. formula_18 being the midpoint of formula_19 yields formula_20. Since formula_21 is the midpoint of formula_22 it is also the midpoint of formula_23, as formula_23 and formula_22 are both diagonals of the parallelogram formula_12. This yields formula_20 and hence formula_28. Therefore, it follows from the intercept theorem (and its converse) that formula_29 and formula_30 are parallel and formula_31, which yields Euler's theorem. Euler's theorem can be extended to a larger set of quadrilaterals, that includes crossed and nonplaner ones. It holds for so called "generalized quadrilaterals", which simply consist of four arbitrary points  in formula_32 connected by edges so that they form a cycle graph.