Geometry with Trigonometry

Sec. 5.6 Area of triangles, and convex quadrilaterals and polygons 73

A

B C

E

F

A

B C

E

F

D

Figure 5.13.

Proof.
The triangles[A,B,E]and[A,C,F]are similar in the correspondence(A,B,E)→
(A,C,F),as∠BAE=∠CAFis in both,|∠AEB|◦=|∠AFC|◦=90, and then by 5.2.2
|∠ABE|◦=|∠ACF|◦. By 5.3.2

|B,E|

|C,F|

=

|A,B|

|C,A|

.

On cross multiplication,

|B,E||C,A|=|C,F||A,B|.

By a similar argument, we can show that|A,D||B,C|is equal to these.
Definition. With the notation of the last result, theareaof the triangle[A,B,C],
denoted byΔ[A,B,C], is the common value of:

1

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

1

2 |B,E||C,A|,

1

2 |C,F||A,B|.

Area of triangles has the following properties:-

(i)If P∈[B,C]is distinct from B and C, then

Δ[A,B,P]+Δ[A,P,C]=Δ[A,B,C].

(ii)If[A,B,C,D]is a convex quadrilateral, then

Δ[A,B,D]+Δ[C,B,D]=Δ[B,C,A]+Δ[D,C,A].

(iii)If two triangles are congruent then their areas are equal.