Sec. 2.4 Triangles and convex quadrilaterals 31
with edgeCAin whichBlies, andH 5 is the closed half-plane with edgeABin
whichClies.However in some courses the actual definition of a triangle is taken to be (i), in
other courses it is taken to be (ii), and in other courses it is taken to be (iii), with (ii)
and (iii) very common. In yet other courses a combination of (i) and (ii) is taken.
Having to make a choice for the sake of precision, we opt for (iii); then for us (i)
will be the set of vertices of our triangle, and (ii) will be the perimeter of our triangle,
with the individual segments being the sides. We shall then be able to refer naturally
to the area of a triangle and the length of its perimeter.
Consideration similar to (i), (ii) and (iii) for a triangle surround each of the terms
quadrilateral, parallelogram, rectangle and square, and we adopt our terminology
consistently.
2.4.2 Triangles................................
NOTE.LetA,B,Cbe points which do not lie on one line. Then by A 1 ,A,B,Care
distinct points, andA∈BC,B∈CA,C∈AB. In fact these lines are not concurrent;
forBCandCAcannot have a pointPin common other thanP=C, whileC∈AB.
Definition. For non-collinear pointsA,B,CletH 1 be the closed half-plane with
edgeBCin whichAlies,H 3 the closed half-plane with edgeCAin whichBlies,
andH 5 the closed half-plane with edgeABin whichClies. Then the intersection
H 1 ∩H 3 ∩H 5 is called atriangle, and is denoted by[A,B,C].
A
B
C
Figure 2.9. A triangle[A,B,C].A
B
D C
Figure 2.10. A convex quadrilateral.The pointsA,B,Care called itsvertices; the segments[B,C],[C,A],[A,B]are called
itssides; the linesBC,CA,ABare called itsside-lines. The union[B,C]∪[C,A]∪[A,B]
of its sides is called itsperimeter. A side and a vertex not contained in it are said to
beopposite; thusAis opposite[B,C]but is not opposite[C,A]or[A,B].
Triangles have the following properties:-(i)[A,B,C]is independent of the order of the points A,B,C.(ii)Each of the vertices A,B,C is an element of[A,B,C].(iii)If P,Q∈[A,B,C],then[P,Q]⊂[A,B,C]so that a triangle is a convex set.