DE LOS CENTROS DE TRABAJO DE PEMEX PETROQUIMICA

Imagine constructing a circle tangent to a given line at a given point. Imagine constructing a second circle, having the same radius as the first, tangent to both the line and the first circle.

Given a circle tangent to a given line, imagine constructing a second circle tangent to the first circle and tangent to the given line at a specified point.

Given a circle tangent to a given line, imagine constructing a second circle tangent to the given line and tangent to the first circle at a specified point.

Comment

You will probably find that when the two circles have equal radius, the construction is not too difficult. Notice how you need to imagine or draw a sketch in order to appreciate what is being asked. The second and third cases require more powerful constructions so that any radius may be chosen for the second circle.Two circles of equal radius will then be a special case of the more general construction.

B O

A C E

D

From all the circle relationships you have discovered earlier, you can probably list many relationships in this diagram. For example, B lies on the line OD; BC is tangent to both circles; CA = CB = CE (equal tangents from a point); the angles at O and D are bisected by OC and DC respectively; DE is perpendicular to CE. Now you can choose which of these properties will help you to construct the figure from a particu- lar starting point.

Suppose you want to construct the second circle to touch the first circle at a given point, like B in Figure 3.4d.The first property listed above says that the centre of the second circle must lie on a line through OB. Focus on B, and look for other elements that could be constructed, knowing B. A few constructions lead to locating D. Similarly, suppose you want to construct the second circle to touch a given point on the line, like E in the diagram.The last property listed above tells you that the centre of the circle must lie on a line from E, perpendicular to the given line. Focusing on E does not yield much else of help, unless you recall the relation between AC and AE. Focusing on C can then suggest a construction to find B and finally D.

The interactive file ‘3d Line and Touching Circles’ provides two constructions along these lines. When you drag E or B, the angle ∠OCD is suggestively invariant, which might direct attention to the four angles at C, how they are equal in pairs and all sum to 180°, showing that the angle is invariant.

This approach to investigation using dynamic geometry software illustrates two very important aspects of the problem. First, you can choose to construct the second circle starting from either position; that is, a point on the circumference of the first circle, or a point on the line. Whichever you choose will become the independent point of your construction. This means that the radius of the second circle will vary according to how you vary this point. Second, the choice of starting point will also determine how the construction proceeds. In other words, you will have two differ- ent constructions depending on your choice.

Two Historical Vignettes

This section concludes with two historical vignettes that provide interesting examples concerning circles. The first is in fact a theorem; the second is an application of circle properties.There are, of course, a vast number of theorems in geometry, some of which are considered to be important enough to be included in most curricula around the world. Probably the first to spring to your mind is Pythagoras’ theorem. Unfortunately, it is also the case that some theorems could almost be described as ‘forgotten’ theorems, in that they rarely appear in secondary school geometry texts. One of these is a powerful theorem derived by Claudius Ptolemy in about AD150. It appears in an astronomical treatise, which later became known as the Almagest, meaning ‘the

greatest’! In fact, Ptolemy is best known for his work in astronomy and trigonometry, and some well-known trigonometric relation- ships can be derived from his geometric theorem.

Using the circle property of equal angles subtended by equal chords, it is not difficult to derive some relationships concerning lengths. Consider the diagram in Figure 3.4e.

Triangles AED and BEC are similar, and so are triangles AEB

and DEC. Using the first of these pairs you can deduce that AE/ED = BE/EC and therefore (AE)×(EC) = (BE)×(ED).You would get the same final result using the other pair of similar triangles, and it is known as the Intersecting Chord Theorem. It 50 BLOCK 1 A E D C B Figure 3.4e

remains true even when the chords intersect outside the circle. However, Ptolemy’s genius enabled him to derive a much more important relationship between the lengths of the cyclic quadrilateral ABCD.

The proof of the theorem depends on deliberately introducing another similar triangle into the diagram. Look at the interactive file ‘3e Ptolemy’ to see how the theorem is derived. As well as deriving trigonometric relationships from the theorem, a simpler, very striking relationship can be quickly established by trying it on a special case. Consider the special case of a rectangle whose sides are of length a and b. Being a rec- tangle it is necessarily cyclic. Now apply Ptolemy’s theorem: the sum of the products of opposite sides is the product of the diagonals. Labelling the lengths of the diagonals as c produces the familiar theorem that lies at the intersection of geometry and alge- bra: Pythagoras! This is a nice example of how different areas of mathematics link together and how one can become a confirming instance of another.

The second historical vignette looks at how knowledge of circle properties can help solve a problem first posed in 1471 by the German astronomer and mathematician known as Regiomontanus.As quoted by Maor (1998, p.46), the problem was expressed in the following way: ‘At what point on the ground does a perpendicularly suspended rod appear largest?’ Put in this way, the problem sounds rather abstract, but it can be put into a real-life context by imagining a statue on a tall column and again asking what is the best position from which to view it so that it appears largest. Gallery-goers can think of a painting high on a wall; rugby players can think of placing the ball for a conversion. In Task 3.4.4, the idea of ‘appears largest’ is re-expressed in a simpler mathematical form.

REASONING BASED ON INVARIANT PROPERTIES 51