BasicQuadrangleIdentity

Basic quadrangle identity

In a quadrangle with sides of fixed length a, b, c, d the expression a*b*cos(x) - c*d*cos(y), where x is the angle of (a,b) and y the angle of (c,d) remains constant and equal to (1/2)(a^2+b^2-c^2-d^2).

a*b*cos(x) is the power of F with respect to the circle with diameter EK. c*d*cos(y) is the power of N with respect to the same circle. Hence the expression is equal to the difference of the squares: (PF)^2 - (PN)^2. This evaluates easily to (1/2)(a^2+b^2-c^2-d^2), using the formula for the medians PF, PN of the triangles EFK and ENK respectively.

The formula for the median: a^2 + b^2 = 2*m + (c^2)/2, where m is the length of the median to side c of a triangle with side-lengths a, b, c. This is proved easily, using Pythagora's theorem.

Look at the file Cyclic_Maximizing.html for a nice application of this fact, concerning a maximizing-property of cyclic quadrangles.

Look at the file BasicQuadrangleIdentity2.html for another identity on quadrangles, pointed out by Michael Metaxas.

http://users.math.uoc.gr/~pamfilos/