All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Shapes > Geometry
Tan2Mid (Posted on 2020-03-11) Difficulty: 3 of 5
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.

See The Solution Submitted by Bractals    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Comment 1 of 1
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
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (2)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2020 by Animus Pactum Consulting. All rights reserved. Privacy Information