Question

Write a proof of your conjecture (the perpendicular bisector of a chord always intersects the midpoint) or give a counterexample.

Answer 1

@VeritasVosLiberabit

Answer 2

write a proof? what level class is this?

Answer 3

Geometry

Answer 4

I would just say

if a and b are two points that lie on a circle. The line segment joining them has a perpendicular bisector that passes through the midpoint.

This is really a proof though and proofs are usually observed in higher mathematics courses . Maybe a mathematician here can give you a simplified version of the proof

Answer 5

this isn't really a proof I meant to say. I don't know how to do a proof for high school geometry

Answer 6

Maybe I can find a proof on wikipedia let me check

Answer 7

Oh okay, thanks.

Answer 8

https://proofwiki.org/wiki/Perpendicular_Bisector_of_Chord_Passes_Through_Center

Answer 9

here is the proof you are looking for. I will try to read it to see if I can simplify it

Answer 10

Join FA,FD,FB.
As F is the center, FA=FB.
Also, as D bisects AB, we have DA=DB.
As FD is common, then from Triangle Side-Side-Side Equality, △ADF=△BDF.

In particular, ∠ADF=∠BDF; both are right angles.
From Book I Definition 10: Right Angle:
When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.
So ∠ADF and ∠BDF are both right angles.
Thus, by definition, F lies on the perpendicular bisector of AB.

Answer 11

This part is the proof

Answer 12

you also need to use the drawing on the wiki page
https://proofwiki.org/wiki/File:BisectorOfChord.png

Answer 13

Thank you sooo much! Proofs always confused me.

Answer 14

This is not typical of any beginning classes. Most high level classes don't really even touch proofs unless you major in mathematics

Answer 15

So don't stress to much if you don't understand it that well.