In the convex quadrangle NPQM, the diagonal NQ is the bisector of the angle PNM and intersects the diagonal PM

In the convex quadrangle NPQM, the diagonal NQ is the bisector of the angle PNM and intersects the diagonal PM at the point S. Find NS if it is known that a circle can be described near the quadrangle NPQM, PQ = 44, SQ = 22

∠QNM – is inscribed in a circle and relies on the arc QM.
∠QPM is also inscribed in a circle and rests on the arc QM.
Therefore, these angles are equal.
∠QNM = ∠QPM
Consider the triangles NPQ and SPQ.
∠SQP – General
∠QNP = ∠SPQ
According to the first sign of the similarity of triangles, these triangles are similar.
Then, NQ / QP = QP / SQ
NQ = QP ^ 2 / SQ = 44 ^ 2/22 = 88
NS = NQ-SQ = 88-22 = 66
Answer: NS = 66