Hello, Krizalid!
I have a rather klunky proof . . .
Prove that two lines are perpendicular if the product of their slopes is
Without trig, vectors . . . just using basic analytic geometry tools.
The first problem is creating a right angle.
I used that fact that an angle inscribed in a semicircle is a right angle. Code:

* * * P
*  o(x,y)
*  o o*
* o o*
o  o
* o  o*
B o+o A
(r,0)*  *(r,0)

*  *
*  *
*  *
* * *

is a point on the semicircle: . .[1]
The slope of is: . .[2]
The slope of is: . .[3]
Substitute [1] into [2]: .
Substitute [1] into [3]: .
Therefore: .