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\parallel AD

Step 2 of 5: Step 2

Previous step Next step

MT=MA,\quad \triangle OMT\sim\triangle ONS

Detailed analysis

When $AB$ and $CD$ meet at $O$ , let $M,N$ be the midpoints of $AB,CD$ , and let $S,T$ be the perpendicular feet from $N,M$ to $AB,CD$ , respectively. The given tangency gives $MT=MA$ , while the right triangles $OMT$ and $ONS$ are similar.