26. Tangents from a point, and two circles

The lesson, step by step

1. Warm up

   Ask: Is a tangent perpendicular to the radius at the point of contact? Why?

2. The new idea

   The tangents to a circle drawn from an external point are equal, and make equal angles with the line joining the point to the centre.

   If two circles intersect each other, the line of centres is perpendicular to their common chord at its middle point.

   If two circles are tangent to each other, the line of centres passes through the point of contact.

3. Draw the figure

   Draw a circle with centre O and a point A outside. Draw the two tangents AB and AC as carefully as you can by eye (an exact construction comes in Prop. XXXVII), and the radii OB and OC.

   Then draw two overlapping circles, their common chord and the line joining their centres.

4. Walk through the proof

   XII: right triangles OAB and OAC share the hypotenuse OA and have equal legs OB and OC (radii), so they are equal.

   XIII: each centre is equally far from both ends of the common chord, so the line of centres is its perpendicular bisector (§ 161).

   XIV: a perpendicular to the common tangent at the point of contact passes through both centres (§ 255).

5. Your student proves it back

   Close the book. Your student redraws the figure from memory, letters it the same way, and gives the proof aloud: what is given, what is to be proved, then each step with its reason ("because the radii are equal", "by § 143"). Point to each line or angle as it is named.

   If a step is missing, ask "How do we know that?" rather than supplying it.

6. Exercises

   Do Ex. 92 and Ex. 95 with drawings: for each case, draw the two circles.

   Ex. 92. Describe the relative position of two circles if the line of centres : (1) is greater than the sum of the radii ; (2) is equal to the sum of the radii ;

7. Wrap up

   Ask: How many common tangents can two circles that cross each other have? (Two.)