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*James King
Univ ersity of Washington, 1994U$lU$lm$lR%
O~kThe special point besides A in the first case is called the line reflection oF A in the line containing the segment. The specialpoint besides K on all the circles in case 2 is called the inversion of the point J in circle 1. Inversions can be constructed directly by straighedge and compass (and Sketchpad) and can be used in many constructions involving orthogonal circles. In each case, the family of circles is called a pencil of circles with two base points.|pl|L A&R X!
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Case 2. Point K on a circle.e.U$m0U$mU !0
9jThese two figures give a way to motivate the idea of reflecting a point across a line or across a circle.
Case1. Point C on a segment. (Scroll down for the second case and then some terminology.)Nmf
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)U~&'MvThe circle through J and K is orthogonal to circle 1. Drag C around circle 1 or double-click the Animate K button to see this "circular" family of circles. How does this compare with the straight family of circles? What is the special point besides J that belongs to this set of circles?
Move J to a new location inside the circle, or on the circle, and repeat the experiment. What does the family of circles look like now?e now?!Action Movement
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Ga)Ru IrThe circle through A and C is orthogonal to segment m. Drag C or double-click the Animate C button to see this "straight" family of circles. What is the other special point on all the circles besides A?
Move A to a new position and repeat the process, observing how the family of circles changes, for example if you drag A onto the segment or to the other side.
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