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While investigating the geometry on the Alpha Triangle thread something unexpected showed up. I personally have never come across this one before, but I would be very surprised if this hasn't turned up somewhere at some time. In fact, I would be amazed if this is an unknown construction.
I am posting it here for interest sake without claim on my part, only that it has turned up in my investigations.

This concerns Isosceles triangles within a circle, actually a circumcircle, a theme I will be developing. This particular part of that theme comes from the triangle legs perpendicular bisectors. The legs can be considered as chords here because only the one leg on one side is shown, for clarity sake.
Here is one chord(leg) and its perpendicular bisector, showing how this goes through the circle's centre. This is an old old trick for finding the centre of a circle, although at least three different chords spread out around the perimeter are normally used for that.

When several chords(legs) are drawn from the same point, the apex of the centre line here, their perpendicular bisectors all go through the circle centre as shown. Any chords could have been chosen with the same result.

When the midpoints of the chords are joined as shown here a pattern emerges

When the perpendicular bisectors of the midpoint connecting lines are added, they too converge at a point, as shown. Again, it doesn't matter which midpoints are chosen for this, any combination would converge at the same point.

The convergent point turns out to be the centre of a circle that is exactly half the size of the original circumcircle, and the midpoints of all chords drawn from the apex of the circumcircle centre line lie on the circumference of the half size circle.

It is this curious development of the midpoints into a circle that I wanted to illustrate here.

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