Math
The inscribed angle and its half
Keep AB fixed. Predict whether the angle will change, then move P and check. The highlighted arc never contains P. Reveal four steps using diameter PD and equal radii. The same construction works by adding or subtracting the small angles; try both cases.
What happens
Keep AB fixed. Predict whether the angle will change, then move P and check. The highlighted arc never contains P. Reveal four steps using diameter PD and equal radii. The same construction works by adding or subtracting the small angles; try both cases. Move P and Q. On the same arc their angles are equal; on opposite arcs they sum to 180°. Coincident points do not make a quadrilateral. The arrow selects the tangent ray corresponding to P. Move P to the other arc: both the angle at P and the selected tangent angle become supplementary. Move A or B until P sees 90°. Then move P to check other positions. The diameter button gives a worked example; overlapping points do not count.
Equations used in the module
Equations α = atan2(|PA × PB|, PA · PB); A ≠ B, P ≠ A, P ≠ B β = arc AB without P, 0° < β < 360°; α = β/2; ε = α − β/2 PD = diameter; OA = OP = OB; ∠AOD = 2x; ∠DOB = 2y α = x+y, β = 2x+2y; or α = |x−y|, β = 2|x−y| AB = diameter ⇒ α = 90° (P ≠ A, P ≠ B) P,Q on opposite arcs: α+γ = 180°; same arc: α = γ tangent ray on the alternate-segment side: τ = α
