Math 487 Lab 11 Part B (Tangents and Circles)

Circle Construction 1

In a new sketch, draw a line AB and a point F.  Also construct a point X on line AB.

Construction Problem:  Construct a circle c with center T that passes through F and is tangent to line AB at X.

Hint:  (1) If the circle passes through F and X then this says that T is on a certain line.  What line?    (2) If the circle is tangent to line AB at point X, then T is on a second particular line.  What line?  (3) If T is on both of these lines, you have constructed T.  Draw the circle through F with this center.

Locus of Centers: Given the construction above, select point T and then point X and choose Construct > Locus.

• Explain why the point T is equidistant from F and line AB.  Conclude that the locus is the set of points equidistant from point F and line AB.
• What is the name of this curve?

A special line

• Move X along line AB.  What relationship does the perpendicular bisector m of FX have with the curve?
• Let p be the line through T perpendicular to line AB (you probably already have constructed this line. Explain why the reflection of a light ray coming along p will be reflected from the tangent line at T to F.  That is why F is called the focus of this curve.
• Construct a locus of lines: Select line m and then point X and choose Construct > Locus.  Drag F around and observe the behavior of this locus.

Circle Construction 2

In a new sketch, draw a circle with center A through point B and a point F inside the circle.  Also construct a point X on circle AB.

Construction Problem:  Construct a circle c with center T that passes through F and is tangent to circle AB at X.

Hint:  (1) If the circle passes through F and X then this says that T is on a certain line.  What line?    (2) If the circle is tangent to circle AB at point X, then T is on a second particular line.  What line?  (3) If T is on both of these lines, you have constructed T.  Draw the circle through F with this center.

Locus of Centers: Given the construction above, select point T and then point X and choose Construct > Locus.

• Explain why the point T is equidistant from F and circle AB.  Conclude that the locus is the set of points equidistant from point F and circle AB.
• What is the name of this curve?

A special line

• Move X along line AB.  What relationship does the perpendicular bisector m of FX have with the curve?
• Check that the line reflection of the point X in this line is point F.  Conclude that a light ray from F will reflect in a tangent line mirror at T to pass through point A.
• Conclude that the sum of the distances from FT + TA is constant.
• Construct a locus of lines: Select line m and then point X and choose Construct > Locus.  Drag F around and observe the behavior of this locus.

A bonus

• Continuing with the same figure, drag the point F outside the circle.  What curve is the locus now?

Circle Construction 3

In a new sketch, draw a circle c and a point A exterior to the circle.

Construction Problem 3A: Construct the lines through A tangent to the circle at points S and T.

(This construction is review, if you need help, look in GTC, Carpenter Chapter, or BG, p. 66.)

Now add a point P to the figure inside angle SAT. 

Construction Problem 3B: Construct two circles, each of which is also tangent to lines AT and AS, but which also pass through P.

Hint:  Draw the line AP.  If M and N are the points of intersection of this line with the circle, then dilate the circle with center A so that M (or N) is dilated to P.

Remark.  You can do this with the Dilate command in Sketchpad.  With straightedge and compass, you will need to use parallels to carry out the construction.  If you draw lines MS and MT and the lines parallel to them through P, you will find the points of tangency with AS and AT of the circle through P.  Likewise the lines NS and NT will give the other circle.

References for Constructions:  The constructions and some other circle constructions are spelled out in GTC, Chapter 6.  Others are in BG.  Another good reference is the Construction Chapter of Birkhoff and Beatley.