Regular polygons and circles

  • So Big
    problem
    Favourite

    So Big

    Age
    16 to 18
    Challenge level

    One side of a triangle is divided into segments of length a and b by the inscribed circle, with radius r. Prove that the area is: abr(a+b)/ab-r^2

  • Orthogonal Circle
    problem
    Favourite

    Orthogonal Circle

    Age
    16 to 18
    Challenge level

    Given any three non intersecting circles in the plane find another circle or straight line which cuts all three circles orthogonally.

  • Area I'n It
    problem

    Area I'n It

    Age
    16 to 18
    Challenge level
    Triangle ABC has altitudes h1, h2 and h3. The radius of the inscribed circle is r, while the radii of the escribed circles are r1, r2 and r3 respectively. Prove: 1/r = 1/h1 + 1/h2 + 1/h3 = 1/r1 + 1/r2 + 1/r3 .
  • Retracircles
    problem

    Retracircles

    Age
    16 to 18
    Challenge level
    Four circles all touch each other and a circumscribing circle. Find the ratios of the radii and prove that joining 3 centres gives a 3-4-5 triangle.
  • Escriptions
    problem

    Escriptions

    Age
    16 to 18
    Challenge level
    For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.
  • Not so little x
    problem

    Not so Little X

    Age
    11 to 14
    Challenge level
    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?
  • Bull's Eye
    problem

    Bull's Eye

    Age
    11 to 14
    Challenge level
    What fractions of the largest circle are the two shaded regions?
  • Shedding Some Light
    problem

    Shedding Some Light

    Age
    7 to 11
    Challenge level
    Make an estimate of how many light fittings you can see. Was your estimate a good one? How can you decide?
  • Triangular Hexagons
    problem

    Triangular Hexagons

    Age
    7 to 11
    Challenge level
    Investigate these hexagons drawn from different sized equilateral triangles.
  • Like a Circle in a Spiral
    problem

    Like a Circle in a Spiral

    Age
    7 to 16
    Challenge level
    A cheap and simple toy with lots of mathematics. Can you interpret the images that are produced? Can you predict the pattern that will be produced using different wheels?
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