Understanding and Constructing Tangencies in Geometry

Tangencies

Making Tangents

Two Circles

• External Tangents: Draw another circle concentric to the larger one with radius R - r. Join the centers. This is the bisector. Draw an auxiliary circle to the centers. Join the center with the points of intersection with the smaller circumference and extend. Where the lines intersect the larger circle will be the tangent points. Parallels are drawn to the smaller circle to find all points of contact.
• Internal Tangents: The process is the same, except that the circle is of radius R + r, and the parallels are drawn to the opposite side.

Tangent to a Circle Passing Through Point P

• If the tangent is to be external, a concentric circle of radius R + r is drawn. From P, an arc of radius R is drawn, which intersects the other circle. The centers of these intersections are the solutions.
• If the tangent is to be internal, the process is similar, but the radius is subtracted instead of added.

Tangent to Both Sides of an Angle

The bisector is drawn, and a parallel is drawn to one side at a distance equal to the radius. The intersection of the bisector and the parallel is the center.

Tangent to Another Circle at a Point on It

Join the center with point T and extend the line. Place the radius on the extension.

Linking Two Circles

• Concave: Draw two concentric circles, adding the radius of the link to each. The intersection points are the centers.
• Convex: The process is similar, but the concentric circles are drawn with the radius of the link minus the radius of the circle.

Tangent to Another Circle at One Point and Passing Through Point P

Join the center with T and extend the line. Join T with P and draw the bisector. The intersection point is the center.

Linking Two Straight Lines

Draw parallels separated by the radius. The intersection point is the center.

Linking Two Lines with Arcs of Different Radii

From the ends of each line, perpendiculars are drawn. A distance greater than that between the respective ends is marked on the perpendiculars. These points are joined, and the bisector is drawn. The first center will be the intersection of the bisector with the outermost perpendicular (considering the outermost to be the one situated further away from the opposite ends of the line). The second center will be the end of the other perpendicular. The junction between the arcs is found by joining the centers and extending that line.

Tangent to a Line and Passing Through Point P

Draw a perpendicular to the line through point T. Join P and T. Draw the bisector of PT. The intersection of the bisector and the perpendicular is the center.

Tangent to a Line and Passing Through Points A and B

Join A and B and draw the bisector. Draw an auxiliary circle passing through A and B. Draw a tangent from point C (where the line intersects the radical axis). The length of the tangent is marked on both sides of C. Perpendiculars are constructed through T1 and T2. The intersections of the perpendiculars with the bisector are the centers.

Circle Tangent to the Sides of an Angle, Passing Through Point P

Draw the bisector. Move P along the bisector to find P'. Join P and P' and extend the line. Draw a circle centered on the bisector passing through P and P'. Proceed as before.

Tangent to a Line and a Circle at a Point on the Circle

Join the center of the circle with T and extend the line. Draw a line perpendicular to the extended line through T, which will intersect the given line. The centers will be on the bisectors of the supplementary angles formed, so they will be at the intersection of these bisectors with the line joining T and the center of the circle.

Tangent to Another Circle Passing Through Points A and B

Join A and B and draw the bisector. Draw an auxiliary circle centered on the bisector and passing through A and B. This circle intersects the given circle. The radical axis of both circles is drawn. From the point where the radical axis intersects the extension of AB, draw tangents to the given circumference. Join the center with T1 and T2, extend the lines, and the intersections with the bisector of AB are the centers.

Circle Tangent to Two Others, at One Point on One of Them

1. Join T with the center and extend the line. Draw a perpendicular to it through T. Draw a circle passing through T that intersects the other circle. Find the radical axis. Where the radical axis intersects the line connecting the centers, draw tangents to the second circle. Proceed as before.
2. Add or subtract the radius of the second circle from T. Join this point with the center of the second circle and draw the bisector. The intersections of the bisector with the line connecting the center with T are the centers.