Cyclic polygons, incircles and lengths in a circle
Build a triangle's circumcircle and incircle and see where their centres are. Then drag the corners of a cyclic quadrilateral round a circle and test two crossing chords: which relationship always stays true?
- UK: Year 10
- Iran: grade 11
- Afghanistan: grade 9
- Syria: grade 7
- Palestine: grade 9
- Ukraine: grade 7
- Russia: grade 8
Guide
For GCSE (Higher circle theorems: opposite angles of a cyclic quadrilateral) and Afghan Grade 9 / Iran Grade 11. The circumcircle, incircle and crossing chords go a little beyond GCSE. The coloured dots can be dragged; the readouts under the figure update live.
A triangle's circumcircle passes through all three vertices. Its centre O is where the perpendicular bisectors of the sides meet, because every point on a perpendicular bisector is the same distance from both ends of the side. In an obtuse triangle O lies outside the triangle.
A triangle's incircle touches all three sides. Its centre I is where the angle bisectors meet, because every point on an angle bisector is the same distance from both arms of the angle. I is always inside the triangle.
In a cyclic quadrilateral (all four vertices on one circle) opposite angles add up to 180°: ∠A + ∠C = 180°. If two chords AB and CD cross at P, then PA × PB = PC × PD.