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So imagine two point A and B.

If we draw a circle C with diameter AB, and have M be a point on this circle, then Angle AMB will always be 90

We can define C as the curve followed by M(x,y) such that Angle AMB = 90. C is then a circle.

But would the curve looks like if we define the angle AMB=30 or 40 or 120?

Would it be an ellipse? A hyperbola? A parabola? Something else?what?

I tried to use geogebra to trace M but I failed, if you can help on that also it would be great!

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    It is an arc. In Spanish it is called "arco capaz", but I don't know how to say it in English. The Wikipedia's page about it has no English version. – 2017-02-19
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    It would be a circular segment, delimited by AB. – 2017-02-19

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It is called major arc segment of a circle or minor arc segment of a circle depending on whether the angle included is acute or obtuse. They can be simply called minor or major arcs. A constant angle is contained in a circle as a geometric property.

Arcs Major/Minor