Interactive resource
Apollonius circle applications
Geometry. On a circle, construct M so k·PB becomes PM; minimize PA + k·PB. Free interactive GeoGebra resource — adjust parameters in the browser.
Overview
On a circle, construct M so k·PB becomes PM; minimize PA + k·PB.
Explanation
When P lies on a fixed circle and you minimize PA + k·PB, build M with PM = k·PB.
Then PA + k·PB = PA + PM — a path from A to the circle to M.
Finish with straight-line/circle geometry or reflection.
How to use
Drag P on the circle; dashed AM meets the circle at the optimum.