Circle properties and circle theorems

  • Crescents and triangles
    problem

    Crescents and triangles

    Age
    14 to 16
    Challenge level
    filled star filled star filled star
    Can you find a relationship between the area of the crescents and the area of the triangle?
  • Similarly so
    problem

    Similarly so

    Age
    14 to 16
    Challenge level
    filled star filled star filled star
    ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.
  • Lens Angle
    problem

    Lens angle

    Age
    14 to 16
    Challenge level
    filled star filled star filled star
    Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.
  • Tricircle
    problem

    Tricircle

    Age
    14 to 16
    Challenge level
    filled star filled star empty star
    The centre of the larger circle is at the midpoint of one side of an equilateral triangle and the circle touches the other two sides of the triangle. A smaller circle touches the larger circle and two sides of the triangle. If the small circle has radius 1 unit find the radius of the larger circle.
  • Three Balls
    problem

    Three balls

    Age
    14 to 16
    Challenge level
    filled star filled star empty star
    A circle has centre O and angle POR = angle QOR. Construct tangents at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q lie inside, or on, or outside this circle?
  • Cyclic Quad Jigsaw
    problem

    Cyclic quad jigsaw

    Age
    14 to 16
    Challenge level
    filled star filled star filled star
    A picture is made by joining five small quadrilaterals together to make a large quadrilateral. Is it possible to draw a similar picture if all the small quadrilaterals are cyclic?
  • Encircling
    problem

    Encircling

    Age
    14 to 16
    Challenge level
    filled star filled star filled star
    An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?
  • Three four five
    problem

    Three four five

    Age
    14 to 16
    Challenge level
    filled star filled star empty star
    Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.
  • Sitting Pretty
    problem

    Sitting pretty

    Age
    14 to 16
    Challenge level
    filled star filled star empty star

    A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

  • Not so little x
    problem

    Not so little x

    Age
    11 to 14
    Challenge level
    filled star empty star empty star
    Two circles are enclosed by a rectangle 12 units by x units. The distance between the centres of the two circles is x/3 units. How big is x?