Let gamma be a circle with center O and P a point outside gamma.

Construct a tangent line to gamma through point P using a straightedge only.

Prove the construction.

Circle γ has its centre O on the line PQ.  Its circumference intersects PQ at B and C  (B is nearer to P).

A line is drawn from P through γ such that D and E form a chord with D being nearer to p.

Draw a line from B through D until a line drawn from C through E intersect.  From this point X construct a line to the centre O.  The intersection of OX with γ is point Y.

Line PY forms the tangent to the given circle.


                 X

                   Y
                             E
              D            

  P          B         O        C          Q

I ask my reader to reconstruct the "imaginary" picture portrayed above.

The closest to a proof that I can give is:
As / DPB widens, D, E and X approach each other, seemingly at point Y which is the apparent tangent intersection.

Edited on March 15, 2008, 8:01 am

Posted by brianjn on 2008-03-15 07:57:18