So Many Intersections

Geometry Level 3

Two non-intersecting circles in the plane \(\Gamma_1\) and \(\Gamma_2\) are drawn centered at points \(O_1\) and \(O_2\) respectively. A circle \(\Gamma_3\) intersects \(\Gamma_1\) at points \(X_1,X_2\) and \(\Gamma_2\) at points \(Y_1,Y_2.\) Another circle \(\Gamma_4\) intersects \(\Gamma_1\) at points \(X_3, X_4\) and \(\Gamma_2\) at points \(Y_3, Y_4.\) Suppose lines \(X_1X_2\) and \(Y_1Y_2\) intersect at \(K_1\) and \(X_3X_4\) and \(Y_3Y_4\) intersect at \(K_2.\) Find \(\angle K_1O_2O_1 + \angle O_2K_1K_2\) in degrees.

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