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Equal touching circles have centres on a line. From a point of this line on a circle, a tangent is drawn to the farthest circle. Find the lengths of chords where the line cuts the other circles.

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!

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The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?

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Draw two circles, each of radius 1 unit, so that each circle goes through the centre of the other one. What is the area of the overlap?

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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.

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If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

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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?

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Three circular medallions fit in a rectangular box. Can you find the radius of the largest one?

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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.

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Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?

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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?

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What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

This is the second article on right-angled triangles whose edge lengths are whole numbers.

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Can you make sense of these three proofs of Pythagoras' Theorem?

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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. . . .

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Which has the greatest area, a circle or a square, inscribed in an isosceles right angle triangle?

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Three squares are drawn on the sides of a triangle ABC. Their areas are respectively 18 000, 20 000 and 26 000 square centimetres. If the outer vertices of the squares are joined, three more. . . .

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Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

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A belt of thin wire, length L, binds together two cylindrical welding rods, whose radii are R and r, by passing all the way around them both. Find L in terms of R and r.

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A ribbon is nailed down with a small amount of slack. What is the largest cube that can pass under the ribbon ?

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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.

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Equal circles can be arranged so that each circle touches four or six others. What percentage of the plane is covered by circles in each packing pattern? ...

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Describe how to construct three circles which have areas in the ratio 1:2:3.

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The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?

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A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be?

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A ladder 3m long rests against a wall with one end a short distance from its base. Between the wall and the base of a ladder is a garden storage box 1m tall and 1m high. What is the maximum distance. . . .

This article is about triangles in which the lengths of the sides and the radii of the inscribed circles are all whole numbers.

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What is the same and what is different about these circle questions? What connections can you make?

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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?

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What remainders do you get when square numbers are divided by 4?

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What are the shortest distances between the centres of opposite faces of a regular solid dodecahedron on the surface and through the middle of the dodecahedron?

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Can you make sense of the three methods to work out the area of the kite in the square?

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A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .

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Is the sum of the squares of two opposite sloping edges of a rectangular based pyramid equal to the sum of the squares of the other two sloping edges?

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Prove that the shaded area of the semicircle is equal to the area of the inner circle.

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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.

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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.

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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.

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Prove Pythagoras' Theorem using enlargements and scale factors.

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The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.

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Draw a square and an arc of a circle and construct the Golden rectangle. Find the value of the Golden Ratio.

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A 10x10x10 cube is made from 27 2x2 cubes with corridors between them. Find the shortest route from one corner to the opposite corner.

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Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.

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The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

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If a ball is rolled into the corner of a room how far is its centre from the corner?

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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.

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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 ?

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Cut off three right angled isosceles triangles to produce a pentagon. With two lines, cut the pentagon into three parts which can be rearranged into another square.

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Can you make a hypothesis to explain these ancient numbers?

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If the altitude of an isosceles triangle is 8 units and the perimeter of the triangle is 32 units.... What is the area of the triangle?