Geometry with Trigonometry

Sec. 5.6 Area of triangles, and convex quadrilaterals and polygons 75

First suppose thatD=B.Then|∠ABC|◦= 90 =|∠A′B′C′|◦and soD′=B′.Then

1

2 |A,D||B,C|=

1

2 |A,B||B,C|=

1

2 |A

′,B′||B′,C′|= 1

2 |A

′,D′||B′,C′|,

so the areas are equal. There is a similar proof whenD=C.
Secondly suppose thatDlies in[B,C]but is notBorC. Then the angles∠CBA=
∠DBAand∠BCA=∠DCAare both acute. It follows that the angles∠C′B′A′and
∠B′C′A′are both acute and soD′lies in[B′,C′]but is notB′orC′.
Now the triangles with vertices{A,B,D}and{A′,B′,D′}have two angles pair-
wise equal in magnitude and so the third angles∠DABand∠D′A′B′are also equal.
Then by the ASA principle of congruence these two triangles are congruent in the
correspondence(A,B,D)→(A′,B′,D′). In particular|A,D|=|A′,D′|and so
1
2 |A,D||B,C|=

1

2 |A

′,D′||B′,C′|. Thus the areas are equal.
Thirdly suppose thatClies in[B,D]but is notD. Then the angles∠CBA=∠DBA
and∠DCAare acute and the angle∠BCAis obtuse. It follows that∠C′B′A′is acute
and∠B′C′A′is obtuse and soC′lies in[B′,D′]but is notD′. Now the triangles with
vertices{A,B,D}and{A′,B′,D′}are congruent as in the second case and so the
equality of areas follows. There is a similar proof whenBlies in[D,C]but is notD,
on interchanging the roles ofBandCin this last argument.

5.6.2 Area of a convex quadrilateral.....................

NOTE.In the quadrilateral in Figure 5.14 we have

Δ[A,B,D]+Δ[C,B,D]={Δ[A,B,T]+Δ[A,T,D]}+{Δ[C,B,T]+Δ[C,T,D]}

={Δ[A,B,T]+Δ[C,B,T]}+{Δ[A,T,D]+Δ[C,T,D]}

=Δ[B,A,C]+Δ[D,A,C].

Definition.Wedefinethe area of the convex quadrilateral[A,B,C,D]as in
Figure 5.14 to be

Δ[A,B,D]+Δ[C,B,D]=Δ[B,A,C]+Δ[D,A,C],

and denote it byΔ[A,B,C,D].

If[A,B,C,D]is a rectangle, then

Δ[A,B,C,D]=|A,B||B,C|,

that is the area is equal to the product of the lengths of two adjacent sides.
Proof.ForΔ[A,B,D]=^12 |A,B||A,D|,Δ[C,B,D]=^12 |D,C||B,C|. As by 5.2.1
|D,C|=|A,B|and|B,C|=|A,D|, the result follows by addition.

5.6.3 Area of a convex polygon

As a preparation for generalisation we first show that in 2.4.4 for the convex quadri-
lateral[A,B,C,D]we haveC∈H 1 ,D∈H 3 ,A∈H 5 ,andB∈H 7 , so that each of the
vertices is in each of these closed half-planes.