Question

Let K be a circle with center P and radius r, and let L be a circle with center O and radius 2r, which passes through P. Let A, B be points on K and C, D points on L, such that lines AC and BD are tangent to both circles. Show that points A, P, D are collinear.

Answer 1

Points A, P, D are collinear. This collinearity is due to the property of tangents drawn from an external point to a circle, resulting in a right angle with the radius of the circle.

Step-by-step explanation:

Consider circle K with center P and radius r, and circle L with center O and radius 2r passing through point P.

Let A and B be points on circle K, and C and D be points on circle L such that lines AC and BD are tangents to both circles.

As AC and BD are tangents to circles K and L respectively, they form right angles with the radii drawn to the points of contact (A and D).

Therefore, angles PAC and PDB are both right angles due to the tangent-circle property.

In a circle, a tangent line is perpendicular to the radius drawn to the point of tangency. Hence, PAC and PDB are 90-degree angles.

By definition, points A, P, and D are collinear if angles PAC and PDB are both 90 degrees.

Therefore, due to the nature of tangents drawn from an external point to a circle forming right angles with the radii at the points of contact, points A, P, and D are collinear.