Other tags that relate to Circles Ad Infinitum
Geometric sequences. Enlargements and scale factors. Pythagoras' theorem. Summation of series. Regular polygons and circles. Visualising. Circle properties and circle theorems. 2D shapes and their properties. Mathematical reasoning & proof. Infinite series.There are 172 results
Broad Topics > Thinking Mathematically > Mathematical reasoning & proof
Fitting In
Age 14 to 16 Challenge Level:
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. . . .
Matter of Scale
Age 14 to 16 Challenge Level:
Prove Pythagoras' Theorem using enlargements and scale factors.
Towering Trapeziums
Age 14 to 16 Challenge Level:
Can you find the areas of the trapezia in this sequence?
Three Balls
Age 14 to 16 Challenge Level:
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?
Long Short
Age 14 to 16 Challenge Level:
What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
Cyclic Quad Jigsaw
Age 14 to 16 Challenge Level:
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?
Zig Zag
Age 14 to 16 Challenge Level:
Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?
Cyclic Quadrilaterals
Age 11 to 16 Challenge Level:
Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?
Pythagoras Proofs
Age 14 to 16 Challenge Level:
Can you make sense of these three proofs of Pythagoras' Theorem?
Similarly So
Age 14 to 16 Challenge Level:
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.
Salinon
Age 14 to 16 Challenge Level:
This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?
Pythagorean Triples I
Age 11 to 16
The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!
Circle Box
Age 14 to 16 Challenge Level:
It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?
Encircling
Age 14 to 16 Challenge Level:
An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?
The Pillar of Chios
Age 14 to 16 Challenge Level:
Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.
Pythagorean Triples II
Age 11 to 16
This is the second article on right-angled triangles whose edge lengths are whole numbers.
Angle Trisection
Age 14 to 16 Challenge Level:
It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.
No Right Angle Here
Age 14 to 16 Challenge Level:
Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.
Parallel Universe
Age 14 to 16 Challenge Level:
An equilateral triangle is constructed on BC. A line QD is drawn, where Q is the midpoint of AC. Prove that AB // QD.
Rhombus in Rectangle
Age 14 to 16 Challenge Level:
Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.
Lens Angle
Age 14 to 16 Challenge Level:
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.
Round and Round
Age 14 to 16 Challenge Level:
Prove that the shaded area of the semicircle is equal to the area of the inner circle.
Pareq Exists
Age 14 to 16 Challenge Level:
Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.
Generally Geometric
Age 16 to 18 Challenge Level:
Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.
Same Length
Age 11 to 16 Challenge Level:
Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?
Folding Fractions
Age 14 to 16 Challenge Level:
What fractions can you divide the diagonal of a square into by simple folding?
Geometric Parabola
Age 14 to 16 Challenge Level:
Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.
L-triominoes
Age 14 to 16 Challenge Level:
L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?
Kite in a Square
Age 14 to 16 Challenge Level:
Can you make sense of the three methods to work out the area of the kite in the square?
Cosines Rule
Age 14 to 16 Challenge Level:
Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.
Picture Story
Age 14 to 16 Challenge Level:
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
More Number Sandwiches
Age 11 to 16 Challenge Level:
When is it impossible to make number sandwiches?
Proof Sorter - Geometric Sequence
Age 16 to 18 Challenge Level:
Can you correctly order the steps in the proof of the formula for the sum of the first n terms in a geometric sequence?
Pent
Age 14 to 18 Challenge Level:
The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.
Triangular Intersection
Age 14 to 16 Short Challenge Level:
What is the largest number of intersection points that a triangle and a quadrilateral can have?
Find the Fake
Age 14 to 16 Challenge Level:
There are 12 identical looking coins, one of which is a fake. The counterfeit coin is of a different weight to the rest. What is the minimum number of weighings needed to locate the fake coin?
Folding Squares
Age 14 to 16 Challenge Level:
The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?
Converse
Age 14 to 16 Challenge Level:
Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?
Picturing Pythagorean Triples
Age 14 to 18
This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.
The Frieze Tree
Age 11 to 16
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
Mouhefanggai
Age 14 to 16
Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.
Mediant Madness
Age 14 to 16 Challenge Level:
Kyle and his teacher disagree about his test score - who is right?
Little and Large
Age 16 to 18 Challenge Level:
A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?
Multiplication Square
Age 14 to 16 Challenge Level:
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Triangle Incircle Iteration
Age 14 to 16 Challenge Level:
Keep constructing triangles in the incircle of the previous triangle. What happens?
Gift of Gems
Age 14 to 16 Challenge Level:
Four jewellers share their stock. Can you work out the relative values of their gems?
Natural Sum
Age 14 to 16 Challenge Level:
The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .
Square Mean
Age 14 to 16 Challenge Level:
Is the mean of the squares of two numbers greater than, or less than, the square of their means?
Proximity
Age 14 to 16 Challenge Level:
We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.
NRICH is part of the family of activities in the Millennium Mathematics Project.