Chord AB is a typical chord passing through a fixed point P in the interior of a circle. Point M is the midpoint of AB and I have drawn in (shown dotted) the two line segments joining P and M to O, the centre of the circle.
By symmetry, the line MO joining the midpoint of the chord to the centre must be perpendicular to the chord. So triangle MOP is right angled, and PO is its hypotenuse.
By the converse of Thales' theorem the hypotenuse of a right angled triangle is the diameter of a circle passing through the three vertices of the triangle, and the centre of the circle is the midpoint of the hypotenuse, N in the diagram above.
The diameter and centre of this circle are fixed by the position of P and O and midpoint M must be on this circle for any chord through P, which is what we wanted to prove.
For information about Thales (fl. 626/623 – c. 548/545 BCE), who seems to have pursued many scientific, mathematical and philosophical interests see here
Proof of interesting geometric fact
This is the proof of the question I posted at https://learn1.open.ac.uk/mod/oublog/viewpost.php?post=285828
Here's a sketch
Chord AB is a typical chord passing through a fixed point P in the interior of a circle. Point M is the midpoint of AB and I have drawn in (shown dotted) the two line segments joining P and M to O, the centre of the circle.
By symmetry, the line MO joining the midpoint of the chord to the centre must be perpendicular to the chord. So triangle MOP is right angled, and PO is its hypotenuse.
By the converse of Thales' theorem the hypotenuse of a right angled triangle is the diameter of a circle passing through the three vertices of the triangle, and the centre of the circle is the midpoint of the hypotenuse, N in the diagram above.
The diameter and centre of this circle are fixed by the position of P and O and midpoint M must be on this circle for any chord through P, which is what we wanted to prove.
For information about Thales (fl. 626/623 – c. 548/545 BCE), who seems to have pursued many scientific, mathematical and philosophical interests see here