Given a circle and two points on that circle,

*P* and

*Q*, draw the chord

*PQ*, and label its midpoint

*M*.

Now draw two other chords of the circle *AB* and *CD* that both pass through *M*.

Further, draw chords *AD* and *BC*.

Label the intersection of *AD* and *PQ*, point *X*.

Label the intersection of *BC* and *PQ*, point *Y*.

_____________________________

Prove that *M* is the midpoint of line segment *XY*.

(In reply to

Let's prove its converse. by Victor Zapana)

Given: An arbitrary circle

Chord PQ, with M as its midpoint

X and Y, such that P-Y-M-X-Q, and YM doesn not equal MX.

Prove: There can be no chords CB and AD, s.t. C-M-D, A-M-B, C-Y-B, and A-X-D at the same time.

Bear in mind, that chords CB and AD must be different, so neither chord can be chord PQ.

Proof: Create an arbitrary chord AD that intersects chord PQ at point X. Thus, A-X-D, for X is in the interior of the circle. This is because P-Y-M-X-Q, and PQ is a chord. Create a chord CD, s.t. C-M-D. Create a chord CB, s.t. C-Y-B. Create line segment AB, which is coincidentally chord AB.

Prove now: M is not on line segment AB.

[To Be Continued