Hello, symmetry!
An interesting problem . . . I had to "invent" my own approach.
I'm certain others have shorter, more elegant proofs.
Prove that if two nonvertical lines have slopes whose product is 1,
then the lines are perpendicular.
Then: .
. . We have two slopes whose product is .
We know that:
So looks like this: Code:
*
* q
* 
*      *
p
And looks like this; Code:
q
*   *

* 
 p
* 

*
Sketch them on the same graph. Code:
 F
 * (p,q)
 a * 
 * 
O *   +   + 
(0,0) 
 * 
 
 b * 
 
 * (q,p)
 G
Now consider the triangle .
Let
Let
Let
We note that: .
. . . and that: .
Since , then is a right triangle with .