MathLabs

Problem 1

Let ABCDABCD be a parallelogram with AB=aAB=a, AD=1AD=1, and ∠BAD=α\angle BAD=\alpha. If △ABD\triangle ABD is acute, prove that the four circles of radius 11 centered at A,B,C,DA,B,C,D cover the parallelogram if and only if a≤cos⁡α+3sin⁡αa\le\cos\alpha+\sqrt3\sin\alpha.
Step 6 of 6: Conclude the equivalence
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a≤cos⁡α+3sin⁡α\boxed{a\le\cos\alpha+\sqrt3\sin\alpha}
Detailed analysis

Combining the reduction, the circumcenter criterion, and the quadratic solution proves the required if-and-only-if statement.