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Geometry

Source: 1993 National High School Mathematics League, Exam Two, Problem 3

February 28, 2020
geometry

Problem Statement

Horizontal line mm passes the center of circle ⊙O\odot O. Line l⊥ml\perp m, ll and mm intersect at MM, and MM is on the right side of OO. Three points A,B,CA,B,C (BB is in the middle) lie on line ll, which are outside the circle, above line mm. AP,BQ,CRAP,BQ,CR are tangent to ⊙O\odot O at P,Q,RP,Q,R. Prove: (a) If ll is tangent to ⊙O\odot O, then AB⋅CR+BC⋅AP=AC⋅BQAB\cdot CR+BC\cdot AP=AC\cdot BQ. (b) If ll and ⊙O\odot O intersect, then AB⋅CR+BC⋅AP<AC⋅BQAB\cdot CR+BC\cdot AP<AC\cdot BQ. (c) If ll and ⊙O\odot O are apart, then AB⋅CR+BC⋅AP>AC⋅BQAB\cdot CR+BC\cdot AP>AC\cdot BQ.
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