# Euler's quadrilateral theorem

Euler's quadrilateral theorem {displaystyle a^{2}+b^{2}+c^{2}+d^{2}=e^{2}+f^{2}+4g^{2}} 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.

Contents 1 Theorem and special cases 2 Alternative formulation and extensions 3 Notes 4 References 5 External links Theorem and special cases For a convex quadrilateral with sides {displaystyle a,b,c,d} , diagonals {displaystyle e} and {displaystyle f} , and {displaystyle g} being the line segment connecting the midpoints of the two diagonals, the following equations holds: {displaystyle a^{2}+b^{2}+c^{2}+d^{2}=e^{2}+f^{2}+4g^{2}} If the quadrilateral is a parallelogram, then the midpoints of the diagonals coincide so that the connecting line segment {displaystyle g} has length 0. In addition the parallel sides are of equal length, hence Euler's theorem reduces to {displaystyle 2a^{2}+2b^{2}=e^{2}+f^{2}} 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: {displaystyle 2a^{2}+2b^{2}=2e^{2}} Dividing by 2 yields the Euler–Pythagoras theorem: {displaystyle a^{2}+b^{2}=e^{2}} 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.[1] Alternative formulation and extensions Euler's theorem with parallelogram 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.