36 CHAPTER 1 Advanced Euclidean Geometry
Theorem.The quadrilateralABCDis cyclic if and
only if
ABĈ +CDÂ = CAB̂ +BCD̂ = 180◦.(1.1)In other words, both pairs of opposite angles add to
180 ◦.
Proof. If the quadrilateral is cyclic, the result follows easily from
the Inscribed Angle theorem. (Draw a picture and check it out!) Con-
versely, assume that the condition holds true. We letCbe circumscribed
circle for the triangle 4 ABC. IfDwere inside this circle, then clearly
we would haveABĈ +CDA >̂ 180 ◦. IfDwere outside this circle, then
ABĈ +CDA <̂ 180 ◦, proving the lemma.
The following is even easier:Theorem. The quadrilateralABCDis cyclic
if and only ifDAĈ =DBĈ.
Proof. The indicated angles open the same arc. The converse is also
(relatively) easy.
Simson’s line (Wallace’s line). There is another line that can be natu-
rally associated with a given triangle 4 ABC, calledSimson’s Line(or
sometimesWallace’s Line), constructed as follows.
Given the triangle 4 ABC, construct the circumcenterC and arbi-
trarily choose a point P on the circle. From P drop perpendiculars
to the lines (BC), (AC),and (AB), calling the points of intersection
X, Y, andZ, as indicated in the figure below.