There are a number of ways to find the coordinates of the intersection of diagonals of a polygon. If your question pertained to a polyhedral, then there is an analogous method.

I. If we know the coordinates of the vertices, we find the intersection of the lines containing the opposite vertices.

Ex. Given quadrilateral ABCD with A(0,7), B(10,5), C(6,-2), and D(-6,3) we find the equation of the line AC and the equation of the line BD and use algebra to find the intersection. (See attachment.)

Slope of AC is -3/2 so the equation is y=-3/2x+7

Slope of BD is 1/8 so the equation is y=1/8x+15/4

Then -3/2x+7=1/8x+15/4

-12x+56=x+30

13x=26

x=2

y=4

So the intersection is (2,4). When solving the system of equations we could use linear combinations, substitution, estimate from the graph and check algebraically, or any number of matrix methods.

II. If we have a special quadrilateral there are some other methods.

a. The intersection of the diagonals of any parallelogram is the midpoint of either diagonal.

b. The intersection of the diagonals of a rhombus additionally lies on the perpendicular bisector of either diagonal.

c. The diagonals of a regular polygon meet at the center (e.g. the point equidistant from all of the vertices, and equidistant from the sides.) The diagonals of a regular 2n-gon (regular polygon with an even number of sides) meet at the midpoints of the diagonals.

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