14 Preliminaries Ch. 1
H 5 is called atriangle. The pointsA,B,Care called itsverticesand the segments
[B,C],[C,A],[A,B]itssides. If a vertex is not the end-point of a side ( e.g. the vertex
Aand the side[B,C]), then the vertex and side are said to beoppositeeach other. We
denote the triangle with verticesA,B,Cby[A,B,C].
If at least two sides of a triangle have equal lengths, then the triangle is called
isosceles.
A
B
C
Figure 1.11. A triangle[A,B,C].A
BDC
An isosceles triangle.- LetA,B,C,Dbe points no three of which are collinear, and such that[A,C]∩
[B,D]=0. For this let/ H 1 be the closed half-plane with edgeAB, containingC,H 3
be the closed half-plane with edgeBC, containingD,H 5 be the closed half-plane
with edgeCD, containingA,H 7 be the closed half-plane with edgeDA, containing
B. Then the intersectionH 1 ∩H 3 ∩H 5 ∩H 7 is called aconvex quadrilateral.
The pointsA,B,C,Dare called its vertices, the segments[A,B],[B,C],[C,D],
[D,A]itssides, and the segments[A,C],[B,D]itsdiagonals. Two vertices which have
a side in common are said to beadjacent, and two vertices which have a diagonal in
common are said to beopposite. ThusAandBare adjacent as they both belong to
[A,B]which is a side;AandCare opposite as they both belong to[A,C]which is a
diagonal.
Two sides which have a vertex in common are said to beadjacent,andtwo
sides which do not have a vertex in common are said to beopposite. Thus the sides
[A,B],[D,A]are adjacent as the vertexAbelongs to both, while the sides[A,B],[C,D]
are opposite as none of the vertices belongs to both of them. The wedge angles∠DAB,
∠ABC,∠BCD,∠CDAare called theanglesof the convex quadrilateral; two of these
angles are said to beadjacentoroppositeaccording as their two vertices are adjacent
or opposite vertices of the convex quadrilateral.
We denote the convex quadrilateral with verticesA,B,C,D, withAandCoppo-
site, by[A,B,C,D].