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# MODULE: Monge's Theorem.

## Part 1: Circles and tangents.

Monge's theorem involves lines which are externally tangent to a pair of circles, so to illustrate it in Sketchpad we must be able to construct such a line. Sketchpad provides no direct way of doing this, or even of directly constructing any tangent line to a single circle. The tools provided are more basic: Sketchpad constructs objects like the line through two points, the circle with a given center and radius, and the line perpendicular to a given one passing through a certain point. Essentially, the tools of Geometer's Sketchpad are the straight-edge and compass, along with some built in constructions from Euclidean geometry. Any other construction should somehow be built in terms of these basic ones. Thus, the first task is to investigate the properties of tangent lines to circles, in order to find such a construction.

Using Sketchpad, construct a circle, a point on it, and a line through this point. Drag the line around, until it appears to become tangent to the circle. Is there a special relationship between the line and the circle? Now draw the segment joining the center of the circle and the point on it. What angle do the two lines seem to form? Measure it using the "measure angle" feature. Is this angle special for the tangent line?

How can we use the previous property to construct a tangent line to a circle? Construct a circle, and a point on it, and then construct the tangent line at that point.

This is all good, but suppose we are given a circle centered at O, and a point Q outside the circle. There are two lines through Q which are tangent to the circle, and we want to be able to construct (one of) them. If we could find which point P on the circle is the point of tangency, then we could construct the line as before. How do we find P? Look at what we know so far: the tangent line should be perpendicular to the radius OP, and it should pass through the given external point Q. Let's look at all the lines satisfying these conditions.

So, construct a circle, and a point Q not on it. Now construct a line through the center O of the circle to some point A, and construct a perpendicular line to it through Q. Let P be the point of intersection of these two perpendicular lines. Now drag point A around, moving the whole configuration, and trace point P. What figure do you seem to get? Make a conjecture. Construct the figure, and check whether your conjecture was right or not.

Of all these possible positions of the points P, which makes line PQ tangent to the circle? Is there more than one? How would you find it (them?) using the tools available in Sketchpad? Putting this all together, given a circle and an exterior point Q, how do you construct the line through Q tangent to the circle? Create a Sketchpad script of this procedure that can then be used again and again, with any circle and exterior point. You may want to hide the figures used in the construction so that only the tangent line appears.

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