Circle properties and circle theorems

  • Circumspection
    problem

    Circumspection

    Age
    14 to 16
    Challenge level
    M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P.
  • Strange Rectangle
    problem

    Strange Rectangle

    Age
    16 to 18
    Challenge level
    ABCD is a rectangle and P, Q, R and S are moveable points on the edges dividing the edges in certain ratios. Strangely PQRS is always a cyclic quadrilateral and you can find the angles.
  • Circumnavigation
    problem

    Circumnavigation

    Age
    14 to 16
    Challenge level
    The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.
  • Flower
    problem

    Flower

    Age
    16 to 18
    Challenge level
    Six circles around a central circle make a flower. Watch the flower as you change the radii in this circle packing. Prove that with the given ratios of the radii the petals touch and fit perfectly.
  • Circle Scaling
    problem

    Circle Scaling

    Age
    14 to 16
    Challenge level
    Describe how to construct three circles which have areas in the ratio 1:2:3.
  • Sports Equipment
    problem

    Sports Equipment

    Age
    7 to 11
    Challenge level

    If these balls are put on a line with each ball touching the one in front and the one behind, which arrangement makes the shortest line of balls?

  • Coins on a Plate
    problem

    Coins on a Plate

    Age
    11 to 14
    Challenge level

    Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.

  • Small pepper seedlings in turquoise pots.
    problem

    Dodecagon Angles

    Age
    11 to 14
    Challenge level

    Weekly Problem 50 - 2012
    The diagram shows a regular dodecagon. What is the size of the marked angle?

  • Subtended angles
    problem

    Circumference Angles

    Age
    11 to 16
    Challenge level

    Can you prove the angle properties described by some of the circle theorems?

  • ArRh!
    problem

    ArRh!

    Age
    14 to 16
    Challenge level

    Triangle ABC is equilateral. D, the midpoint of BC, is the centre of the semi-circle whose radius is R which touches AB and AC, as well as a smaller circle with radius r which also touches AB and AC. What is the value of r/R?

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