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 4 of 6: Translate coverage into a quadratic
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a2−2acos⁡α+1−4sin⁡2α≤0a^2-2a\cos\alpha+1-4\sin^2\alpha\le0
Detailed analysis

Substituting the expression for RR into R≤1R\le1 and using sin⁡α>0\sin\alpha>0 yields exactly the displayed quadratic inequality.