Four short experiments on the circle — drag the points yourself. Every theorem in this lab is something you can watch stay true, no matter how hard you try to break it.
Every point, the same distance
Definition
Drag the gold point P — the circle redraws itself through it. Now drag the green point Q; notice it can slide anywhere but is always forced back onto the circle. That's the definition: a circle is just every point at one fixed distance from a centre.
drag P to resize the circle · drag Q around it
Readings
Distance C → P (radius)—
Distance C → Q—
Are they equal?—
CP and CQ always match, however you drag them — that equality is the circle.
Chords and their distance from the centre
Theorems 5–8
Drag the two gold endpoints of chord AB along the circle. Watch the perpendicular distance d from centre C, and check the length formula AB = 2 √(r² − d²). A second fixed chord (grey) is drawn for comparison — the longer chord is always the one closer to the centre.
drag either gold endpoint of chord AB
Readings
Radius r—
Distance d (centre → AB)—
Chord length AB (measured)—
2√(r² − d²) (formula)—
Grey chord CD length—
Closer to centre?—
Make AB longer than CD — see which one hugs the centre.
The centre sees double
Theorem 9
A, B are fixed on the circle; drag point D anywhere else on the circle, outside arc AB. The angle ∠ACB at the centre C (shaded red) is always exactly double the angle ∠ADB seen from D (shaded green) — for every position of D on the major arc.
drag A, B or D — all three move freely on the circle
Readings
Central angle ∠ACB (arc AB)—
Inscribed angle ∠ADB—
Ratio central ÷ inscribed—
Drag D closer to A or B — the ratio holds even at the extremes.
Opposite corners, straight angle
Theorem 11
Drag any of the four vertices A, B, C, D around the circle. Because all four sit on one circle, opposite angles of quadrilateral ABCD always add to 180° — try to drag them into a shape that breaks it.