Question 4
Two circles with centers $O,O'$ and radii $R,R'$ respectively meet at two points $P,Q$. A variable line $\Delta$ is always cut by the two circles in a harmonic range of four points.
(a) From $O,O'$ drop perpendiculars $OH,O'H'$ to $\Delta$. Use logical reasoning to conjecture the loci of $H,H'$ as precisely as possible.
(b) Prove those conjectures and determine the limits of the loci. In what case does the locus pass through the two points $O,O'$?
(c) Assuming $OO'\lt \sqrt{R^2+R'^2}$, find a point $M$ on $\Delta$ for which the sum of the lengths $MO$ and $MO'$ is the shortest among all paths joining $O$ to $O'$ through a point on $\Delta$. Prove this shortest length is constant as $\Delta$ varies. How should the question be changed when $OO'\gt \sqrt{R^2+R'^2}$?
Convention: the two intersection points belonging to each circle form one of the two harmonic pairs; all four points must be distinct real intersections. Tangencies are excluded. The source does not specify whether “the locus” means its full supporting circle or its allowed arcs; the solution distinguishes them.
(a) From $O,O'$ drop perpendiculars $OH,O'H'$ to $\Delta$. Use logical reasoning to conjecture the loci of $H,H'$ as precisely as possible.
(b) Prove those conjectures and determine the limits of the loci. In what case does the locus pass through the two points $O,O'$?
(c) Assuming $OO'\lt \sqrt{R^2+R'^2}$, find a point $M$ on $\Delta$ for which the sum of the lengths $MO$ and $MO'$ is the shortest among all paths joining $O$ to $O'$ through a point on $\Delta$. Prove this shortest length is constant as $\Delta$ varies. How should the question be changed when $OO'\gt \sqrt{R^2+R'^2}$?
Convention: the two intersection points belonging to each circle form one of the two harmonic pairs; all four points must be distinct real intersections. Tangencies are excluded. The source does not specify whether “the locus” means its full supporting circle or its allowed arcs; the solution distinguishes them.