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Brahmagupta's theorem can easily be extended.
Extension of Brahmagupta's Theorem
If a cyclic crossed quadrilateral is orthodiagonal (that is, two of the lines passing through the 4 vertices and not coinciding with the sides intersect perpendicularly at a point M outside the circle), then the perpendicular to a side from the point M always bisects the opposite side (intersecting sides are opposite each other).
Image: https://ibb.co/1X00bxH
In this Geogebra app you can move the yellow control point outside as well as inside the circle where Brahmagupta's theorem applies: https://www.geogebra.org/m/wthe5qt2