This presentation describes how to find the common internal tangents of two circles. It is a 7 step process: 1) Join the centers of the circles. 2) Draw the perpendicular bisector of the line between centers. 3) Draw a circle centered at the midpoint with a radius to one center. 4) Add the radii to get the radius of an auxiliary circle. 5) Find the intersection points of this circle with the first circle. 6) Draw lines from the first center through these points to find tangent points, and draw parallels from the second center. 7) Join the tangent points to obtain the common internal tangents in blue.

1. This presentation has been made for you by
FRANCISCO GUIJARRO BELDA
From Isaac Albéniz Secondary School
Leganés. Madrid. Spain.

2. COMMON INTERNAL TANGENTS TO TWO GIVEN CIRCLES
1) Let s consider we already have the two circles, O1 and O2. The first step to
take is to join both centers, in red color in the image

3. 2) Draw the perpendicular bisector of O1O2 and you will get M, the middle
point of this segment

4. 3) Draw from M a circle (in green color in the image) with radius MO1

5. 4) Set aside segment r1 (in the upper side of the image below), radius of the circle O1.
Place next to it segment r2, radius of the circle O2. You will get r1+r2 (in red color),
which is going to become the radius of a new auxiliary circle. Keep in mind that this is
the addition of the two radius involved

6. 5) Draw a circle with center in O1 and radius r1+r2. You will get two intersection points
in the green circle, Ta and Tb

7. 6) Join point O1 to Ta , as a consequence, this line intersects the black circle in T1.
Repeat the process with O1 and Tb in order to get T2. Draw lines parallel to these ones
from O2 and you will get O2T3 and O2T4. Remember, O1T1 is parallel to O2T3 and O1T2
to O2T4

8. 7) In order to finish, you only have to join point T1 to T3 and T2 to T4. You have the
solutions in blue color in the image below