Edit: Hmm ... this seemed shorter in my head. Please bear the long-windedness of this post ...
Let the point be arbitrarily labeled .
Now if you form , it is essentially composed of 3 line segments:
Let be the respective slopes of .
We have: ..... .....
Now, we're told that the sums of the slopes equal 1, i.e. . Since , we have:
Also, note that is the same thing as saying that . If you remember, in order for two line segments to be perpendicular to one another, their slope one of line segment must be the negative reciprocal of the other.
So in order for , we must have:
Equate the red and blue to get:
Notice that they're reciprocals of each other. The only value that is equal to its own reciprocal is 1 and -1.
So either: .....or.....
But we're told that the sum of the slopes is equal 1 (from above in magenta):
Do you see how this can never happen?