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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OBOA=OCOD⟺BC∥AD\frac{OB}{OA}=\frac{OC}{OD}\Longleftrightarrow BC\parallel AD
Detailed analysis

The second tangency is equivalent to OB/OA=OC/ODOB/OA=OC/OD. By the converse intercept theorem, this ratio equality is equivalent to BC∥ADBC\parallel AD.