Inscribe Circles in Triangles Using Geometric Construction

A slide show of experiments with interactive geometry software.

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Definition of Inscribed Figure

In geometry, an inscribed planar shape or solidis one that is enclosed by and "fits snugly"inside another geometric shape or solid.

Specifically, at all points where figures meet,their edges must lie tangent. There must beno object similar to the inscribed object but

larger and also enclosed by the outer figure.

From Wikipedia, the free encyclopediahttp://en.wikipedia.org/wiki/Inscribed_figure

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Prerequisites

It will be helpful if you are already familiar with:

•Pythagoras’ theory of right angled triangles.•Compass and straight edge construction.•Inscribing a circle in a triangle.•Transformations in the Euclidean Plane

If not, look them up after the slide show.

Goal

This topic should encourage you to take a fresh look at familiar shapes, to ask questions, to do experiments with geometry and mathematics, and to prove your results.

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Motivation

Interesting constructions can be formed from circles inscribed in an isosceles right triangles, as noted in reference [1]. What other shapes that are worth investigating? An isosceles right triangle is half of a square. What interesting constructions can we create from a circle inscribed in a half of an equilateral triangle? Let's experiment, using interactive geometry software to draw the constructions. The next slide compares an isosceles and half of an equilateral triangle, each with inscribed circles.

[1] Inscribe Semicircle in Squarehttp://www.slideshare.net/cmcallister/inscribe-semicircle-in-square-by-geometric-construction

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Interactive Geometry Software

Interactive geometry software provides compass and straightedge construction, and additional tools such as the midpoint of a line, and parallel or perpendicular lines.

The free Dr Geo software (by OFSET) was used for this slide show. You can use any interactive geometry software, or simply a pair of compasses and a ruler.

The next slide is representation of Pythagoras’ theory, created using Dr Geo. The constructed triangle is half of an equilateral triangle. It is a right angled triangle. The square of the hypotenuse is equal to the sum of the squares of the other two sides. This equality is can be seen from the area of the three squares in the diagram.

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Experiment

Let's experiment, using interactive geometry software to draw the geometric constructions.

Use a triangle which is half of an equilateral triangle. The ratio of its height to its base is the square root of three, by Pythagoras’ theorem.

A set of different sized triangles can be drawn by using the side of one triangle as the hypotenuse of the next.

Experiment with a variety of triangles, lines and circles.Look for patterns, symmetry and geometric coincidences, for example an unexpected intersection, tangent or square.

The next slide shows some brainstorming with geometry.

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Recursion

A set of triangles of decreasing size can be drawn by using the side of one triangle as the hypotenuse of the next.

The triangles are rotated in steps of 30 degrees, becoming smaller in each turn. Inscribe a circle in each triangle.

Rotate the triangle eight times to produce nine circles.

CalculateThe ratio of side lengths between adjacent triangles.The ratio of diameters between one circle and the next.The ratio of areas between one circle and the next.The area of the ninth circle relative to the first circle.

Suggest a useful application of this series of circles.Suggest an easier way to draw a spiral of triangles.

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Transformation of Square

The triangles are rotated in steps of 30 degrees, becoming smaller on each turn.

The yellow and red colouring highlights similarities over two turns, which is 60 degrees.

Three triangles form the partial boundary of a square. The circle in the second triangle is in the centre of the square.

The second square is rotated 60 degrees and its side is a fraction of the length of the side of the first square.What fraction?

An intermediate square at 30 degrees has been omitted from the drawing. Can you see it?

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Rotational Symmetry

Take the second triangle from the previous slide, and extend it to the base of the square to form an equilateral triangle.Inscribe three circles in the triangle. This construction is unchanged by a 120 degree rotation about its centre.

It is evident that: Three equal circles can be inscribed in an equilateral triangle, and each circle is in the centre of a square of which a side of the triangle is a side of the square.

ExerciseAdd more circles to the diagram.

Where is the centre of each circle you added? What points does it go through?

Why?Prove it!

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Division of a Right Angle

Beginning with half of an equilateral triangle, add more triangles with the side of the first used as the hypotenuse of the next. Notice that the 30 degree angles of the yellow, green and grey triangles add up to form a right angle.

The triangles decrease in size, and their short sides connect to form an approximate spiral.

Inscribe a circle in each triangle. Notice that the line through the centre circle and through the right angle divides the right angle into two equal angles.

What do you notice about the line through the centres of the circles inscribed in the yellow and green triangles?

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Isosceles in Equilateral Triangle

Continuing from the previous construction, the lines sloped at 45 degrees can be emphasised by forming the right isosceles triangle, shown in red. One side of the red triangle is defined by the centres of the circles inscribed in the yellow and green triangles. A grey square can be added that has the same base as the red isosceles triangle.

Do you notice any other coincidences in this diagram?

Can you explain why an equilateral triangle inscribed with circles contains a right isosceles triangle?

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An Abstraction of Electricity

This is an abstract representation of three-phase electricity. Electricity is useful, powerful and potentially dangerous. You might be surprised that mathematics is critical for its study.

Compass and straightedge can be used to draw the “Y” and “Delta” of three-phase electricity. The “Y” shape is shown as blue lines and the “Delta” as a red equilateral triangle.Can you find a hexagon in the diagram?Prove that the Delta is the largest triangle in the circle.

What is the ratio of lengths of the red and blue lines?How does the Pythagorean Theorem (slide 8) relate to this representation of three phase electricity (slide 18)?

The blue “Y” also represents the three cube roots of 1, in the complex plane. (Look up Argand Diagram & Roots of Unity.)

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Credits

This slide show and included geometricconstructions are in the public domain.

Constructions drawn using Dr. Geo software.Geometry files uploaded to:http://i2geo.net/ by colinmca

Slideshow and constructions byColin McAllister, blogging at:

http://cmcallister.typepad.com/