Constructing an accurate tangent line along the exterior of a circle

I am using Geometer's Sketchpad version 5.04 and I wanted to illustrate the the law that:

1. If a secant segment and a tangent segment to the same circle share an endpoint in the exterior of the circle, then the square of the length of the tangent segment equals the product of the lengths of the secant segment and it's external segment (AP)^2 = (PC)(PB)

I attached my attempt. I was not able to get a nice accurate intersection between segment and circle to form a tangent that would illustrate this rule. The secants were fine, I think the error was in making a tangent. Thanks for your help.

You will need to construct the tangent line to make it stay tangent to the circle (as opposed to drawing the line to intersect the circle). One way to do this is to construct a line perpendicular to the endpoint of a radius of the circle. (Construct the radius of the circle, select the radius and its endpoint on the circle, and choose Construct | Perpendicular Line.)

I'm attaching an example sketch. I picked a point on the tangent line to be the exterior point and then constructed the segment from that point to the point of tangency to create the tangent segment.

Constructing an accurate tangent line through a point exterior to the circle

Hi Rob,
Elizabeth's answer is a very good one, provided you're willing to start your construction using the point of tangency, and go from there.

But if you want to start your construction with the exterior point and then construct both the secant and the tangent, it's much more challenging. As you observe, constructing the secant is easy but constructing the tangent is hard. In fact, this is one of my very favorite geometric challenges. I love it because it is easily expressed ("from a point P exterior to circle AB, construct a line tangent to the circle") but doing the construction turns out to be very challenging. I don't want to give away any answers to such a nice challenge; that would take too much of the fun out of it. So let me instead provide a hint that may help you both to figure out a solution and at the same time figure out why it works.

Of course, as with any good challenge, I implore the reader: don't look at the hint until you've struggled with the problem for a while. Only after you've struggled should you read the next paragraph.

Because this problem is hard, it's useful to try an easier related construction with the same given objects: circle AB and exterior point P. As an experiment, construct another independent exterior point Q, and line PQ. Leaving points A, B, and P fixed (because they're the given point for the desired construction), drag Q around P to view all possible lines through point P. Now drag point Q so that line PQ is fairly close to being a tangent line, but far enough away that the line doesn't actually touch the circle. How can you construct a point on line PQ that would be the point of tangency if line PQ were actually a tangent? (I won't give this question away either, but I will say that it's easier than the main question, and after you construct this point you can test your construction by dragging Q and seeing that your constructed point on line PQ becomes the point of tangency when line PQ is tangent to the circle.)

Now that you've found this point -- the point that would be the point of tangency if PQ were really a tangent line -- turn on tracing for it, and drag point Q again to see the pattern formed by the traced point. Analyzing this pattern can help you answer the original construction challenge, and also to explain why your resulting construction must be correct.

I hope I've described this hint clearly enough to make it useful, without spoiling your fun in solving the challenge.

--Scott

P.S. If you get stuck on the second, "easier" challenge, think about the relationship between the tangent line, the center of the circle, and the point of tangency.

I thank you both very much! Elizabeth, I tried what you said -- create a radius segment and construct perpendicular lines! Duhhhhh! I also tested that my tangent line was 90 degrees from the radius and it was.

By the way, what do you do when you take the marker and try to create an angle marker and it just won't do it? Sometimes it absolutely refuses, I either get nothing, or I get what looks like a splotch of lipstick (the marker)

Stek, I ended up taking the tan line close to circle and thought, what the heck, I'll make another circle and see what happens. The circle touched the other circle so it was clear that was an intersection. I constructed a radius of the little tiny circle, then clicked both points of the radius (and maybe the radius itself, it was too small to see) and I constructed parallel lines.

Is that the solution? I attached my resulting .gsp file, called "circle_relationships_tangent" I hope.

When I took a look at your sketch, I couldn't find an independent point, exterior to the circle, that determined a tangent line. So I think that my text-based description was not clear enough.

I've now made a video (available on our Sine of the Times blog as Challenge #2) in which I explain the question more clearly, and also give a hint that may be helpful yet not give too much away.

I often find that a text-based explanation of dynamic geometry constructions is inadequate; it fails to embody either the dynamic or the visual qualities of the construction I'm trying to explain. So I hope the video is helpful!

So here's what I came up with. I made a segment along the trace line, and then put a point on the circle in that segment, and then made a perpendicular line. Is that it?

Constructing an accurate tangent line along the exterior of a circle

Hi Rob,

You're on the right track, with the perpendicularity between the tangent line and the extended radius in your sketch. That relationship of perpendicularity is at the heart of what it means to be tangent. But you haven't labeled the exterior point P in your sketch, so I don't know which of the points is the "given" point through which the tangent is supposed to go. I've enclosed a picture of the construction up to (but not including) the crucial step: it has the given circle, the given point P, and an independent point Q that you can use to explore all possible lines through P. So first, construct and label a sketch that matches the picture.

The next step, the crucial one, is to figure out how to construct another point T on line PQ that will be the point of tangency when Q is dragged to make the line PQ tangent to the circle. Think about the critical relationship of perpendicularity as you try to figure out a construction that will locate the correct point T on line PQ. (By the way, my video intentionally hid the construction line that I used to construct point T; I didn't want to give away too much.)

If you get desperate, send me your email address and I'll send you privately a sketch with the completed construction.

OK Scott, so I made a perpendicular line to the PQ line, and then made an intersection that was point T and I can drag point T up and down the perpendicular line. However, it won't make an intersection, it won't "attach" to the circle. How do you make it stick to the circle?

SPOILER ALERT: This post contains a more detailed hint

So here's a more detailed hint that will be of use if you've gotten frustrated trying to follow my original (minimal) hint. Given the circle with center C and the point P exterior to the circle, and after constructing line PQ so that you can drag Q to easily explore all possible lines through P, you can think about the relationship between line PQ and the circle when the line is tangent, and realize that when this is the case, the intersection T of the perpendicular through point C to line PQ is the point of tangency. So go ahead and construct that perpendicular and point T, and trace point T as you move point Q. In this way you can observe the pattern formed by T, which will give you a very nice hint about an approach to actually constructing T.