Finding the tangent lines between a point and a circle isn’t too hard if you’re familiar with the example Determine where two circles intersect in C#.

Consider the figure on the right. R is the radius of the circle. You can easily calculate the distance D between the external point and the circle’s center by using the Pythagorean theorem. If the point P is (Px, Py) and the circle’s center C is (Cx, Cy), then .

The tangent meets the circle’s radius at a 90 degree angle so you can use the Pythagorean theorem again to find .

Believe it or not, you’re now done because the tangent points P0 and P1 are the the points of intersection between the original circle and the circle with center P and radius L. Simply use the code from the example Determine where two circles intersect in C# to find those points.

The following code shows how the `FindTangents` method used by the example program finds the tangent points.

// Find the tangent points for this circle and external point. // Return true if we find the tangents, false if the point is // inside the circle. private bool FindTangents(PointF center, float radius, PointF external_point, out PointF pt1, out PointF pt2) { // Find the distance squared from the // external point to the circle's center. double dx = center.X - external_point.X; double dy = center.Y - external_point.Y; double D_squared = dx * dx + dy * dy; if (D_squared < radius * radius) { pt1 = new PointF(-1, -1); pt2 = new PointF(-1, -1); return false; } // Find the distance from the external point // to the tangent points. double L = Math.Sqrt(D_squared - radius * radius); // Find the points of intersection between // the original circle and the circle with // center external_point and radius dist. FindCircleCircleIntersections( center.X, center.Y, radius, external_point.X, external_point.Y, (float)L, out pt1, out pt2); return true; }

The code calculates the distance D squared. It uses that to calculate L and then calls `FindCircleCircleIntersections` to find the intersections between the two circles. See the previous example for a description of the `FindCircleCircleIntersections` method.

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