On the Areas of Triangles Associated with an Excircle

S

1

≈

4.78

S

2

≈

3.24

S

3

≈

1.54

S

2

+

S

3

≈

4.78

Let ABC be a triangle. Let D and E be the points of contact of the extensions of AB and AC with the excircle opposite A. Let I be the incenter of ABC. let G and F be the intersections of DI and EI with BC. Let

S

1

,

S

2

, and

S

3

be the areas of GIF, DBG, and ECF, respectively. Then

S

1

=

S

2

+

S

3

.