Let P be a point outside a circle with center O.
Let A and B be the two points on the circle such that
lines PA and PB are tangent to the circle.
Let N be an arbitrary point on line AB.
Let the line through N and perpendicular to ON
intersect lines PA and PB at points C and D respectively.
Prove that N is the midpoint of CD.
Consider the circle with diameter OD. B and C are both on this circle because DBO and DNO are right angles. Therefore inscribed angles ODN and OBN are equal.
Similar reasoning on the circle with diameter OC gives equal angles OCN and OAN.
Now since triangle AOB is isosceles, so is triangle COD. So altitude ON bisects the base CD. N is the midpoint of CD.
Posted by Jer
on 2020-03-14 22:48:23