Assignment Seven: Tangents

Rekha Payor

A line can touch a circle 0, 1, or 2 times. If the line does not touch the circle, the circle and the line do not intersect each other. If the line touches the circle once, it is said to be tangent. If a line touches a circle twice, then the line cuts the circle.

A special property of the tangent is that if a radius of the circle is draw to the point of tangency, the tangent and this radius will be perpendicular. This is proved by Euclid in Book III Proposition 18. The proof is as follows:

Given a circle ABC with center F and line DE tangent to the circle at point C. Want FC to be perpendicular to DE.

Proof by Contradiction: Assume FC is not perpendicular to DE. So construct a line from point F that is perpendicular to DE (Proposition I.12). Then since angle FGC is right by construction, angle FCG is acute (Proposition I.17). But we know the great angle subtends the greater side, so line FC, which is subtended by angle FGC, is greater than line FG, which is subtended by angle FCG (Proposition I.19). Notice line FC is equal to line FB for they are both radii of the circle. This means that line FB is also greater than line FG, but this is impossible since line FB and line BG make up line FG. Hence, FG is not perpendicular to DE. Similarly, we can show that no other straight line besides FC is perpendicular to DE.

Now we can construct a circle that is tangent to two circles. This can be done using the following script tool in GPS. Below is a picture of the script tool. The dotted circle is the circle that is tangent to both green circles. Notice the red ellipse that was created by tracing the center of the tangent circle as the highlighted point at the top of the picture was animated.

Two circles will intersect when the following inequality holds, where r1 is the radius of the smaller circle and r2 is the radius of the larger circle (note: if the circles are the same size then either circle can be r1 and r2):

r2-r1<distance between the centers<r1+r2

Given this information and the fact that circles can cross either 0, 1, or 2 times, we can examine the location of the centers of the constructed tangent circles given the circles intersect. Let the circles intersect at 2 points. Then the following diagram can be constructed (again the dotted circle is the circle that is tangent to both green circles and the red ellipse was created by tracing the center of the tangent circle as the highlighted point at the top of the picture was animated). Notice when the circle intersect, the tangent circle can no longer be draw inside the larger circle.

Let the circles intersect at 1 point, with the little circle inside the larger one. Then the following picture was created (the same notions are used). Here we can see a distinct red ellipse that was formed.

Return