Pythagoras' theorem

  • Retracircles
    problem

    Retracircles

    Age
    16 to 18
    Challenge level
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    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.
  • 30-60-90 Polypuzzle
    problem

    30-60-90 polypuzzle

    Age
    16 to 18
    Challenge level
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    Re-arrange the pieces of the puzzle to form a rectangle and then to form an equilateral triangle. Calculate the angles and lengths.
  • Square World
    problem

    Square world

    Age
    16 to 18
    Challenge level
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    P is a point inside a square ABCD such that PA= 1, PB = 2 and PC = 3. How big is angle APB ?
  • Get Cross
    problem

    Get cross

    Age
    14 to 16
    Challenge level
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    A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?
  • Strange Rectangle
    problem

    Strange rectangle

    Age
    16 to 18
    Challenge level
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    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.
  • Reach for Polydron
    problem

    Reach for polydron

    Age
    16 to 18
    Challenge level
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    A tetrahedron has two identical equilateral triangles faces, of side length 1 unit. The other two faces are right angled isosceles triangles. Find the exact volume of the tetrahedron.
  • Fitting In
    problem

    Fitting in

    Age
    14 to 16
    Challenge level
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    The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ
  • Three cubes
    problem

    Three cubes

    Age
    14 to 16
    Challenge level
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    Can you work out the dimensions of the three cubes?
  • Orthogonal Circle
    problem

    Orthogonal circle

    Age
    16 to 18
    Challenge level
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    Given any three non intersecting circles in the plane find another circle or straight line which cuts all three circles orthogonally.
  • Six Discs
    problem

    Six discs

    Age
    14 to 16
    Challenge level
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    Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?