Let PQ be a diameter of a circle,
     A be a point on line PQ such that P lies between A and Q,
     T be a point on the circle such that line AT is tangent to the circle,
     B be the point on line QT such that line BP is perpendicular to line PQ, and
     C be the point on line PT such that line CQ is perpendicular to line PQ.

Prove that points A, B, and C are collinear.

I'll be the first to admit that geometry is the area of math that I am the weakest at, so take this counter-proof with a grain of salt :-)

Now since PTQ subtends a diameter then angle PTQ is a right angle.  Since PTC is a straight line then angle QTC is also right.  Also since QTB is a straight line then angle BTC is also right.

Now for the part I'm not 100% on, if ABC is a straight line and since ABQ is a right angle then CBT would also need to be a right angle, but then that would mean that triangle CBT would have 2 right angles and that obviously can't happen so A,B,C and not be collinear.

I really can't see where my flaw is in this, but like I said my geometry knowledge is not the greatest so I anticipate someone finding a flaw in my counterproof, if not then this reminds me of the kind of devious questions my high school geometry teacher would put on tests where she would ask us to proof something that isn't true, expecting us to see that and give a counter proof. :-)

Posted by Daniel on 2009-12-12 10:37:21