Orthogonal Circles

1. What does "orthogonal" mean in general?
2. What is the angle between two curves and how is it measured?
3. When are a line and a circle orthogonal?
4. When are two circles orthogonal?
5. What are the relations among distances, tangents and radii of two orthogonal circles?
6. Given circle c with center O and point A outside c, construct the circle d orthogonal to c with A the center of d
7. Given points A and B on c, construct circle d orthogonal to c through A and B.
8. Given point A on c and B not on c, construct circle d orthogonal to c through A and B.

1. What does "orthogonal" mean in general?

The Greek roots for the word are "ortho" meaning right (cf. orthodox) and "gonal" meaning angle (cf. polygon and polygonal). Two geometrical objects are orthogonal if they meet at right angles.

2. What is the angle between two curves and how is it measured?

The angle between two curves is defined at points where they intersect. The angle at such as point of intersection is defined as the angle between the two tangent lines (actually this gives a pair of supplementary angles, just as it does for two lines. The angle may be different at different points of intersection.

This is a natural definition because a curve and its tangent appear approximately the same when one zooms in (i.e., dilates ths figure), as shown in these figures. Follow this link to Zooming in on the Tangents for figures showing this.

Reference: GTC, 9.1, p. 147-8.

Examples to consider

• Sketch two curves that intersect at a point P; then slide your ruler to approximate the tangents. Then measure the angle between them with a protractor.
• Draw two lines that intersect at a point Q and then sketch two curves that have these two lines as tangents at Q.
• If two curves are tangent at P, what is the angle between the two curves?
• How can you measure the angle between a line and a curve that intersect at P?
• Draw two circles that intersect at P. How can the tangents be constructed exactly?
• Draw two lines that intersect at a point Q. How can one construct two circles through Q with these tangent lines?

3. When are a line and a circle orthogonal?

Construct an example of a circle and a line that intersect at 90 degrees.

• How is the line related to the center of the circle?
• Given a circle c with center O and a point A, how can you construct a line through A that is orthogonal to c?

4. When are two circles orthogonal?

Construct an example of two circles that intersect at 90 degrees at a point T.

• How are the two tangent lines at T related to the centers of the circles?

5. What are the relations among distances, tangents and radii of two orthogonal circles?

Suppose c is a circle with center P and radius r and d is a circle with center Q and radius s. If the circles are orthogonal at a point of intersection T, then angle PTQ is a right angle.

• Since T is on c, |PT| = r.
• Since T is on d, |QT| = s.
• Since angle PTQ is a right angle, PQ is the hypotenuse of the right triangle PTQ and |PQ|² = r² + s².

Constructions

6. Given circle c with center O and point A outside c, construct the circle d orthogonal to c with A the center of d.

Draw the figure with c and A. The key to this construction is to recognize that the tangents to P through c are diameters of d.

7. Given points A and B on c, construct circle d orthogonal to c through A and B.

Reference: GTC, 9.1, p. 150.

8. Given point A on c and B not on c, construct circle d orthogonal to c through A and B.

Reference: GTC, 9.1, p. 152.