Consider the triangle defined by $ A(3, -5)$, $B(1, -3) $ and $ C(2, -2) $. I need to get the length of the angle B external bisector. Here my picture
I think I know the process:
- Find the distances
- Get the ratio using the exterior bisector theorem
- Use the ratio formula to get abscissa and ordinate
- And finally get the BD distance
The distances:
$ AB = {\sqrt 8},\quad BC = {\sqrt 2} $
The ratio:
$ \frac{AB}{BC} = \frac{DA}{DC},\quad \frac{\sqrt 8}{\sqrt 2} = \frac{DA}{DC}\quad \frac{DA}{DC} = 2 $
But I'm struggling with the ratio. If I change $ \frac{AB}{BC} = \frac{DA}{DC} $ to $ \frac{AB}{BC} = \frac{AD}{CD} $, I will get the exact same ratio, but different differences in the ratio formula: $$ r = \frac {x - x_1}{x_2 - x}$$
Ok, that's the expected thing. But the question is:
What is the correct order for the bisector theorem when using it with ordered pairs?
Maybe from left to right, or from the bigger side to the short one? I don't know.
PD: I would like to stick with this approach since I'm still practicing with the very basics, as you can see :D And the main question is the order for the bisector theorem, not the bisector distance actually.
$2 = \frac {x-3}{x-2}\\ 2x - 4 = x- 3\\ x = 1$