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**Introduction to Tangent to Circles**

In Euclidean plane geometry, tangent lines to circles form the subject of numerous theorems, and perform an important role in several geometrical constructions and proofs. Since, tangent to a circle at a point P is perpendicular to the radius to that point; theorems including tangent lines often include radial lines and orthogonal circles.

**Tangent lines to one circle:**

A tangent line t to a circle C intersects the circle at a particular point T. For assessment, secant lines intersect a circle at two points, while another line may not intersect a circle at all. This property of tangent lines is conserved under many geometrical transformations, such as scaling, rotation, translations, inversions and map projections. If we say in technical terms, these transformations do not change the incidence structure of the tangent line and circle, although the line and circle may be malformed.

**Tangent lines to two circles:**

For two circles, there are generally four bitangent lines that are tangent to both. For both of them, the external tangent lines, the circles fall on the similar side of the line; for the others, the internal tangent lines, the circles fall on contradictory sides of the line. The external tangent lines traverse in the external homothetic center, while the internal tangent lines traverse at the internal homothetic center. The internal homothetic center constantly lies on the line joining the centers of the two circles, in the segment among both circles. The exterior center lies on the identical line and is nearer to the center of the smaller than to the center of the larger. If both of the circles have identical radius, the external tangent lines are parallel and - at least in inversive geometry - the external homothetic center, slander at infinity.

** Belt problem**

The internal and external tangent lines are useful in solving the belt problem that calculates the length of a belt or rope needed to fit snugly over two pulleys. If the belt is measured to be a mathematical line of insignificant thickness and if both pulleys are supposed to lie in exactly the similar plane, the problem transfers to summing the lengths of the relevant tangent line segments with the lengths of circular arcs subtended by the belt. If the belt is draped about the wheels so as to cross, the interior tangent line segments are related. On the other hand, if the belt is draped exteriorly across the pulleys, the exterior tangent line segments are related; this case is sometimes called the pulley problem.

**Illustration**

Generally the points of tangency t1 and t2 for the four lines tangent to two circles with centers x1 and x2 and radii r1 and r2 are given by solving the simultaneous equations:

**Outer tangent**

The red line joining the points (x3, y3) and (x4, y4) is the outer tangent among the two circles. Specified points (x1,y1), (x2,y2) the points (x3,y3), (x4,y4) can easily be calculated by equating the angle theta and adding the x, y-coordinates of the triangle(theta) to the original coordinates (x1,y1) as shown in the figure.

**Inner tangent**

An inner tangent is a tangent that intersects the segment joining two centers of the circles. This should be noted that the inner tangent will not be defined for cases when the two circles overlap.

**Monge's theorem**

Monge's theorem is named after Gaspard Monge in geometry which states that for any three circles in a plane, no one of which is inside one of the others, the three intersection points of the three pairs of external tangent lines are in fact collinear.

Monge's theorem lets us know that surprisingly the three "points" of the "ice-cream cones" are always in a straight line. In the visual example underneath the red, green, and blue points are all in a straight line.

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