Circumscribed and inscribed circles of triangles and regular polygons
MathematicsPlane GeometryAges 14–15
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Plane Geometry
Drag the three vertices of a triangle: the circumcenter O is the intersection of the three perpendicular bisectors, the incenter I is the intersection of the three angle bisectors; the radii R and r and special cases (right triangle: O is the midpoint of the hypotenuse; equilateral triangle: O coincides with I) update instantly. Regular-polygon mode: change the number of sides n and drag a vertex to change the radius, revealing how R, r and side length a relate to each other. New: medians and the centroid G (ratio 2 : 1), altitudes and the orthocentre H, the Euler line, the midsegment and its converse, an isosceles lock with equal base angles, and a quadrilateral mode for Varignon's theorem.
Lesson: Circumscribed and inscribed circles of a triangle; regular polygons
What it shows
Every triangle has a circumcircle, centred at O where the perpendicular bisectors meet (R = abc/(4S)), and an incircle, centred at I where the angle bisectors meet (r = S/p). The medians meet at the centroid G, which divides each median in the ratio 2 : 1; the altitudes meet at the orthocentre H; and O, G and H lie on the Euler line with GH = 2·OG. A midsegment is parallel to the third side and half as long. An isosceles triangle has equal base angles. The midpoints of any quadrilateral form a parallelogram (Varignon). In a regular n-gon both circles share one centre.
How to use
Drag vertices A, B, C. Toggle Perp. bisectors and Angle bisectors to see which lines meet at O and which at I, then tick Medians, Altitudes, Euler line, Midsegment or Isosceles (CA = CB) and read the ratios under the drawing; untick Circles for a cleaner view. Open Tasks for three challenges. Regular polygon compares R, r and the side a; Quadrilateral shows Varignon's parallelogram.
Parameters you can change
- Shape Triangle (drag the three vertices), Regular n-gon, Quadrilateral (Varignon midpoints)
- Number of sides n of the regular polygon 3–12
- Show perpendicular bisectors
- Show angle bisectors
- Show circumcircle and incircle
- Show medians and centroid G
- Show altitudes and orthocentre H
- Show Euler line (O, G, H)
- Show midsegment C′B′
- Isosceles triangle (CA = CB)
Questions to explore
- When does the circumcentre lie outside the triangle, and where is it for a right triangle?
- For an equilateral triangle with side a, how many times larger is R than r?
- As the number of sides n increases, how does the ratio r/R change, and what shape does the polygon approach?
