Part 2 2
Part 2
Part
cMaking your own tools
by Michael Fox
Last time, I discussed the pros and cons of using geometrical
Now select the line-segment and the midpoint and choose
software. Now it's time to do some practical work with
Construct, Perpendicular Line; with the keyboard you can use
Sketchpad: these notes apply to version 4 - there is a free
Alt+C, then D (from Perpendicular).
upgrade that you can download to convert it to version 4.07.
In this article we shall look at making what Sketchpad calls
I want this tool to give only the two starting points, A and B,
and their mediator, not the segment AB or its midpoint C.
Custom Tools. Other programs call them macros; but they do
the same task: they carry out complete constructions
So select the points A and B and the mediator. Make sure C
and the line-segment AB aren't highlighted.
automatically.
I shall assume that you already have at least a basic
knowledge of how to use Sketchpad, such as you might
C teClick to select
acquire with a few hours' use. You should be able to
Click to select
construct diagrams, not necessarily very complicated; you
Click to select
ought to be familiar with using the menus, and know how to
select objects drawn on the screen. Remember that you can
choose your own labels for points: select a point, press
AB
Alt+/, type the new label in the box that appears, and click
B
on OK.
You need to have the Toolbox on your screen. This is a set
of six square buttons showing a pointer, a point, a line, a
circle, an A, and two black triangles. If it's not there, go into
the Display menu, and look-f6r Show Toolbox. Click on that,
Fig. 1
and the toolbox will appear. You can drag it around by
There is another way, which is sometimes easier for an
moving the cursor on to any part of it that's not a button,
then clicking and holding the left mouse button to drag the
elaborate diagram: hide (but do not delete) the objects you
toolbox to where you want it. Mine is parked on the extreme
don't want to appear. Then select what is left. So hide the
left of the screen, and is a vertical column with the six
segment and the midpoint: select them and use Display,
buttons. Three of the buttons have a tiny triangle pointer in
Hide Objects, or press Ctrl + H. (If you delete them instead of
the bottom right-hand corner. If you click and hold on these
hiding them, the mediator will disappear as well. There will
buttons, you get some alternatives. So the line button gives
be times when you make this mistake; to retrieve what has
you access to a line-segment (which is finite), a ray (going to
gone, go immediately to Edit, Undo, or press Ctrl + Z.) With
infinity in one direction), or a line (which in theory goes off
just two points and a line left on the screen it's easy enough
to infinity in both directions).
to select them by clicking on each, but with more elements
it's quicker to use Edit, Select All, or Ctrl+A.
Now we can start. Let's begin with a very simple tool:
constructing the mediator of two points. It's often called the
perpendicular bisector (of the line-segment joining the
points), but that's eight (or 17) syllables instead of four. I'll
go through this in some detail. We shall speed up later.
This has set everything up; so we now make the tool. Click
on the Custom Tool button - with the two triangles - and
choose Create New Tool. In the New Tool box that appears
type Mediator as the tool name. You can see the steps in the
construction if you tick the Script View box, and click OK.
Using the line-segment tool (the button with a line having
no arrows at its ends), click in two different places on the
screen. This gives two points and the line-segment joining
them. With the line-segment selected, construct its mid-
The script box can be enlarged: drag its sides outwards, and
you can confirm that the construction requires two points to
be given. Close the Script View by clicking on the cross at the
top right.
point. You can use the mouse, choosing Construct then
Midpoint, but it's quicker to press Ctrl+M; this does the job
immediately.
6
To use the tool, click again on the custom tool button, and
put two points on the screen. The mediator follows as you
Mathematics in School, November 2008 The MA web site www.m-a.org.uk
move the cursor to place the second point. If the two points
Try making a tool for the three escribed circles. Another
are already there, just click on each in turn.
possible tool would give the medians and centroid of a
triangle; or you could try the altitudes and orthocentre.
A tool is no good if you can't use it when you want to. You
can delete the two points and their mediator and the tool
You may have wondered why I have not saved the sides of
will still be there. You can open a new sketch without closing
the triangle as part of these tools. It's quite easy to include
the original one, and the tool can be used in it; simply press
them, but generally you will have drawn them anyway, so
the Custom Tool button and select Mediator. But if you close
there is no need for the tool to do it as well.
the original sketch without saving it, then the tool is lost.
There's not much point in making tools unless we are going
All tools are stored in the Tool Folder, which is within the
to use them, so let's look at a few things that can be done.
main Sketchpad folder. If there is no Tool Folder, you must
I am a great believer in going beyond the boundaries of
create one. Then with the sketch that has the construction,
the syllabus, showing students that there is more to
choose Save As... , go to the Tool Folder, and save your
mathematics than is in textbooks. Sometimes we can prepare
document as My Tools. When you start Sketchpad again, click
the way for later work; certainly we can show in simple
the Custom Tool button, and in the list you should see My
ways how investigation plays a part in mathematical
discoveries.
Tools. Move the cursor down to this, and a box showing
Mediator opens. Click on it to use it.
Investigation 1
If you want to add more tools, go to File, Open (or Ctrl+O) and
find your way to Tool Folder. Click on it to open it, and load My
Draw a triangle ABC using lines, not line segments, and put
Tools. Create any new tools you want, then File, Save (or
points D, E, F on its sides, as in Fig. 4.
Ctrl + S). If you can't find a tool that you know you have saved,
just open My Tools and keep it on the screen. You can then use
any of the tools in it in any sketch that you have on screen.
A
Try this by making a tool that draws a circle through three
given points. Put three points A, B, C on the screen. Use
your Mediator tool first with A and B, then with A and C,
and put a point where the mediators cross.
S
BiA
B-- D C
Hide
Fig. 4
B1 C
With the circumcircle tool put a circle through A, E, F; one
through B, F, D; and one through C, D, E. What happens?
Does this property still hold if you move D, E or F along
their respective lines? You can experiment by moving D,
say, on to the extension of BC. Can you prove this property?
(Hint: let the first two circles cut at S. Use angles associated
Fig. 2
with the cyclic quadrilaterals AESF and BFSD to prove
something about CDSE.)
Select this point, then the point A, and Construct, Circle by
Center + Point (or Alt+C, then C). Hide the mediators, go to
Now put a point A' on the circle AEF, and draw AA'.
Create New Tool, and name it Circumcircle.
Construct parallel lines through B and C, meeting the
corresponding circles in B' and C'. Draw the line A'B'. What
Make another tool for Incircle, using the diagram as a guide.
happens? Why?
(To bisect the angle ABC, select A, B and C in that order,
then Construct, Angle bisector.) When you have finished the
Drag A' round its circle. What happens? Is it what you
construction, hide everything except the three starting
expected? My diagram broke down for a time: B' stuck when
points, the circle and its centre. Then make the tool, calling
it reached B, then freed itself - yours may do the same
it Incircle.
(Fig. 5). This occurs because Sketchpad has two points of
A
intersection to choose from, and after B' coincides with B it
selects the wrong one.
2
1
To prevent this happening, we can force the correct choice.
Instead of constructing a parallel line through B, construct a
63
line perpendicular to AA', through the centre of circle BFD.
Then use this as a mirror in which to reflect B. You do this
by either double clicking on the line, or selecting it and
B5C
using Transform, Mark Mirror (or Alt+T, Alt+M). Then
select B and choose Transform, Reflect (or the equivalent
4
Alt+T, Alt+F). This gives a unique B', and there is no
choice for Sketchpad to make (Fig. 6). Use a similar
Fig. 3
Mathematics in School, November 2008 The MA web site www.m-a.org.uk
7
Investigation 3
Draw a circle and take four points A, B, C, D on it. Join them
to form the cyclic quadrilateral ABCD, and draw its two
diagonals. Use your Incircle tool to construct the incircles of
ABC, BCD, CDA, DAB (Fig. 8).
S
E
A
DC
D
Fig. 5
procedure to find C'. It is always worth using this process
B
when you need a particular intersection of a line and a circle.
C
Fig. 8
Find their radii, and look for a simple combination of them
that always gives the answer zero. (The proof of this is quite
difficult.) Join the centres of the circles to form a new
quadrilateral. Does it appear to be special?
B~ D C
Investigation 4
Draw four lines that meet each other in distinct points. If we
left out each line in turn we would make four different
triangles. Draw the circumcircles of all four. What happens?
Fig. 6
Construct a circle through any three of the circumcentres.
Investigation 2
What happens?
Draw a triangle ABC and find the midpoints of the sides.
There is a point where several circles meet. Draw
Label them D, E, F. Construct the centroid G. Construct the
perpendiculars to the four lines from this point, and mark
altitudes and the orthocentre, H. Use your Circumcircle tool
where each perpendicular cuts the corresponding line,
to draw a circle with centre O through A, B, C, and one with
giving four new points. Can you say anything about these
centre N through D, E, F (Fig. 7). What do you notice about
points?
these centres, O and N ?
Construct the orthocentres of the four triangles. Can you say
anything about them? Can you see a possible relationship
between these four points and the previous four? (Here
again the proofs, which go back to the nineteenth century,
are quite hard.)
Solutions
BF
Last time I left you with two problems. The first was about
a chain of circles in a pentagon with an axis of symmetry.
Let the pentagon be ABCDE, and let us say that a circle
touching the sides that meet at A is inscribed in A. Inscribe a
circle in A, then put one in B touching the A circle, then one
in C touching the B-circle, and so on. Will you eventually
reach a circle inscribed in E that also touches the first circle?
Fig. 7
There is a simple solution. Imagine that a long chain of
By selecting each circle in turn and using the Measure menu,
circles has been constructed. The axis of symmetry must
you can find their radii. What is the relationship between
pass though a vertex (and through the midpoint of the
them? What other points does circle DEF pass through?
opposite side, which will be perpendicular to it). Take that
Where does it cut the line segments AH, BH and CH? Can
vertex as A and start with any circle inscribed in it, such as
you prove your conjectures? (This circle is the 'nine-point
1 in the diagram (Fig. 9).
circle'. It was discovered at the end of the eighteenth
century.)
8
Mathematics in School, November 2008 The MA web site www.m-a.org.uk
there are four inside and four outside C. If you try this, it's a
ED
good plan to draw the construction lines in a pale colour,
2a
and hide them after each circle is found, otherwise your
5
diagram will look like a tangle of knitting needles. My
version of this is Figure 11.
-3 4
12,
3
BC
Fig. 9
Note that its centre must be on the axis of symmetry. Now
work both ways along the chain. The circles 2 and 2a in B
and E are mirror images in the axis. So are 3 and 3a in C and
D. Going on round, the next D and C circles (4 and 4a)
match each other, as do 5 and 5a. The new circle in A that
touches the E circle will be the mirror image of the new one
touching the B circle. But they both have their centres on
the axis of symmetry, so the only possibility is that they are
Fig. 11
the same circle, so the chain of circles is closed.
Keywords: Constructions; Dynamic geometry software; Sketchpad.
Author
Michael Fox
e-mail: [email protected]
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MATHEMATICAL ASSOCIATION
Its a Kind
/m
Of Magic
Fig. 10
uppor n m.em
Coming
Soon
The second problem was to construct a circle touching two
This book contains many numerical
given lines I and m, and a given circle C with centre O. (This
tricks, suitable for classroom
is quite a hard problem.)
enrichment or general
mathematical entertainment,
The usual way of devising a construction is to suppose that
together with full mathematical
somehow it has been done, and see what properties the
explanations as to why the tricks
actually work. There is also a
finished diagram has that could be used in constructing it.
section dealing with mathematical
Suppose that the two circles touch at a point Q. Then there
card tricks that require no special
is an enlargement with centre Q that turns one circle into
card-handling skills to perform but
the other, and it would turn each given line into a parallel
still have the potential to create a
line touching circle C. So draw lines touching C and parallel
wow-factor amongst any observers.
to I and m. To do this, draw perpendiculars to I and m that
I hope you enjoy using these tricks
pass through O, then draw tangents through the points
as much as I have enjoyed compiling them. David Crawford
where these perpendiculars cut C. This gives a parallelogram
surrounding C. Draw a line n though a vertex and the
To reserve your copy, telephone sales on
intersection of I and m. Suppose n cuts C at Q. The diagram
o1i6 2210014 or email [email protected]
shows one of the two possibilities. Then OQ cuts an angle
bisector of I and m in the centre of the required circle, which
The Mathematical Association
must pass through Q. This always works unless O and Q are
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on the angle bisector. In this case draw a line through Q
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tangent to C, and extend it to make a triangle with I and m.
The required circle touches the three lines and so is the
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incircle (or an escribed circle) of this triangle.
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If the intersection of the two lines is inside C, you should be
able to construct eight circles that touch C and the two lines;
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9