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# MODULE: Monge's Theorem.

## Part 1: Circles and tangents.

When trying to illustrate the construction for Monge's theorem, we have to construct tangent lines to several pairs of circles. SketchPad provides no direct way of doing this, not even a direct construction of a tangent line to a single circle. The tools provided are more basic; SketchPad allows only constructions such as a line through two points, a circle with a given center and radius, and a line perpendicular to another one passing through a certain point. See the instructions on SketchPad for a more complete list. Any other construction should somehow be phrased in terms of these basic ones. The first task is then, to investigate the properties of tangent lines to circles.

Using SketchPad, construct a circle, construct 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, a point on it, and construct the tangent line at that point.

This is all good, but still we don't have a way of constructing a tangent line passing through a point outside the circle. We would know how to do it, once we find which point on the circle is the point of tangency. How do we find it? Look at what we know so far: the tangent line should be perpendicular to the radius, and it should pass through the given external point, Q, say. Let's look at all the lines satisfying these conditions.

Again, construct a circle, and a point Q not on it. Now construct a pair of lines: one through the center O of the circle, the other through the point Q, that are perpendicular to each other. Let P be the point of intersection of the two lines. Now move these lines around (by dragging point A, the point (other than O) defining one of the lines), 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 the possible positions of the points P, which one gives you a point of tangency? Is there more than one? How would you find it (them?) using the tools available in Sketchpad? And how would you then construct the tangent lines to a circle through an exterior point? Can you create a Sketchpad script of this procedure that can then be used again and again, with any circle and exterior point?

Next: Tangents to circles