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AEduMath 25 (12/2007)RQSBXPCYFigure 7As ΔBPS ~ ΔCPY and ΔBSX ~ ΔCYS , we haveBP SB SX .PC CY SYAR BP CQ AS SX SYHence 1RB PC QA SX SY AS.RemarksThe “Usual Pro<strong>of</strong>” is adopted from Geometry Revisited <strong>by</strong> Coxeter andGreitzer (1967). Some <strong>of</strong> the students’ pro<strong>of</strong>s appear also in some geometrytextbooks. For example, pro<strong>of</strong>s similar to “Pro<strong>of</strong> 1” and “Pro<strong>of</strong> 2” are alsopresented in Posamentier’s book Advanced Euclidean Geometry (2002).One common point <strong>of</strong> all the pro<strong>of</strong>s <strong>by</strong> the students is the use <strong>of</strong> auxiliaryparallel lines on the triangle to relate ratios <strong>of</strong> segments to other ratios <strong>of</strong>segments. During the workshop, I encouraged the students to first assumesome numerical values on the ratios AR : RB and CQ : QA and try tocalculate the remaining ratio BP : PC with the help <strong>of</strong> adding auxiliary81
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