Inscribed Angles

An inscribed angle has its vertex on the circle and its two sides as chords reaching out to an arc. The central angle for that same arc has its vertex at the center. The inscribed angle is always exactly half the central angle, which means it's half the arc measure too, since arc measure is defined to match its central angle. Drag the inscribed angle's vertex anywhere else on the remaining part of the circle and it still opens onto the same arc, so it stays exactly the same size. Only the position of the vertex changes, not the angle.

The proof uses two radii drawn from the center to the endpoints of the chords and to the vertex, splitting the figure into isosceles triangles. Each triangle has two equal radius sides, so its base angles match. Adding up the triangle angles shows the inscribed angle's measure is always half the arc's central angle, whether the center sits inside the angle or outside it. That's why pulling the vertex closer to the arc never makes the angle bigger: closeness doesn't matter, only which arc the angle intercepts.

This idea connects directly to chords: any two inscribed angles that open onto the same chord from opposite arcs are supplementary, since together they intercept the full circle, 360 degrees, split so each piece's inscribed angle covers half its arc, and the two arcs add to 360. It also sets up tangent-chord angles and cyclic quadrilaterals, where opposite angles are supplementary for the same reason.