MathLabs

Problem 4

Let ABCD be a convex quadrilateral such that CD is tangent to the circle with diameter AB. Prove that AB is tangent to the circle with diameter CD if and only if BC∥ADBC\parallel AD.
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OMMT=ONNS⟹OBOA=ON−NSON+NS\frac{OM}{MT}=\frac{ON}{NS}\Longrightarrow\frac{OB}{OA}=\frac{ON-NS}{ON+NS}
Detailed analysis

Similarity gives OM/MT=ON/NSOM/MT=ON/NS. Using the convex ordering along the two lines and MT=MAMT=MA, this becomes OB/OA=(ON−NS)/(ON+NS)OB/OA=(ON-NS)/(ON+NS).