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I want to find the intersection point of two line segments determined by the position of four locators. 

l11 = {0, 0}; l12 = {4, 4}; l21 = {0, 4}; l22 = {4, 0}; l1 = Line[{Dynamic[l11], Dynamic[l12]}]; l2 = Line[{Dynamic[l21], Dynamic[l22]}]; Graphics[{Locator[Dynamic[l11]], l1, Locator[Dynamic[l12]], Locator[Dynamic[l21]], l2, Locator[Dynamic[l22]], PointSize[3], RegionIntersection[{l1, l2}]}, PlotRange -> 10, Axes -> True]

Mathematica gives an error:

Line[{{0, 0}, {4, 4}}] is not a correctly specified region

As far as I understood, the error is caused by my Line expressions containing Dynamic expressions.

How can I fix my code to show the intersection of two such line segments?

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You can fix your code by rewriting it only in terms of the four points where the locators are positioned.

DynamicModule[{l11 = {0., 0.}, l12 = {4., 4.}, l21 = {0., 4.}, l22 = {4., 0.}}, Dynamic @ Graphics[ {Locator[Dynamic[l11]], Line[{l11, l12}], Locator[Dynamic[l12]], Locator[Dynamic[l21]], Line[{l21, l22}], Locator[Dynamic[l22]], Red, AbsolutePointSize[8], RegionIntersection[{Line[{l11, l12}], Line[{l21, l22}]}]}, PlotRange -> 10, Axes -> True]]

intersect

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