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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AB∥CD⟺AB=CDAB\parallel CD\Longleftrightarrow AB=CD
Detailed analysis

If ABAB and CDCD are parallel, then ABAB is tangent to the circle with diameter CDCD exactly when AB=CDAB=CD. In a convex quadrilateral this is equivalent to ABCDABCD being a parallelogram, hence to BC∥ADBC\parallel AD.