Transcription of Cyclic Quadrilaterals | The Big Picture
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Cyclic Quadrilaterals | The Big Picture
Winter Camp 2009 Cyclic QuadrilateralsYufei ZhaoCyclic Quadrilaterals The Big PictureYufei important skill of an olympiad geometer is being able to recognize known , many geometry problems are built on a few common themes. In this lecture, we will exploreone such What Do These Problems Have in Common?1. (IMO 1985) A circle with centerOpasses through the verticesAandCof triangleABCand intersects segmentsABandBCagain at distinct pointsKandN, respectively. Thecircumcircles of trianglesABCandKBNintersects at exactly two distinct that OMB= 90 .ACBKNMO2. (Russia 1995; Romanian TST 1996; Iran 1997) Consider a circle with diameterABand centerO, and letCandDbe two points on this circle. The lineCDmeets the lineABat a pointMsatisfyingMB < MAandMD < MC. LetKbe the point of intersection (different fromO) of the circumcircles of trianglesAOCandDOB. Show that MKO= 90 .ABCDOMK3. (USA TST 2007) TriangleABCis inscribed in circle . The tangent lines to atBandCmeet atT.
Winter Camp 2009 Cyclic Quadrilaterals Yufei Zhao Cyclic Quadrilaterals | The Big Picture Yufei Zhao yufeiz@mit.edu An important skill of an olympiad geometer is being able to recognize known con gurations.
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