SECTION 1.3 Circle Geometry 37
Theorem. The points X, Y, and
Z, constructed as above are colin-
ear. The resulting line is called
Simson’s line(orWallace’s line)
of the triangle 4 ABC.
Proof. Referring to the diagram we note that PZB̂ andPXB” are
both right angles. This implies thatXPẐ +ZBX̂ = 180◦ and so the
quadrilateralPXBZ is cyclic. As a result, we conclude thatPXZ” =
PBẐ. Likewise, the smaller quadrilateral PXCY is cyclic and so
PCÂ =PCŶ =PXY”. Therefore,
PXZ” = PBẐ
= PBÂ
= PCÂ (angles open the same arc)
= PCŶ
= PXY” ,
which clearly implies thatX,Y, andZ are coliner.
Ptolemy’s Theorem.If the quadri-
lateral ABCD is cyclic, then the
product of the two diagonals is equal
to the sum of the products of the op-
poside side lengths:
AC·BD=AB·CD+AD·BC.
When the quadrilateral is not cyclic, then