CircleBundleLocus

Circle-Bundle locus

Consider a member-circle (c) of the bundle (I) of circles orthogonal to a given circle (b) and a, non-intersecting (b), line (a) (see CircleBundles.html ). Then
1) The common chord KL of the two circles passes through a fixed point O, independent of (c).
2) The middle M of KL describes a circle d(N, |NM|), N being the middle of OA, A the center of (b).
3) The symmetric J of P w.r. to M describes a circle f(H, |HJ|) passing through B.

The proofs are easy:
1) |OK||OL| = |OP||OB| = |OQ||OR| = const., Q and R being the limiting points of bundle (I) .
2) angle(AML) is a right one etc.
3) e is the homothetic of d w.r. to P. It passes through B, since JBH and MAN are homothetic isosceli.
By the way, J is the orthocenter of triangle BKL, which is similar to BSC, whose orthocenter is always P. See Polar.html for the whole story.

One can fix circle c and vary circle b, asking for the locus of the orthocenter P. See the file Polar2.html for the relevant picture.

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