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Hyperbola (Locus Construction)

In this diagram, Line p is the perpendicular bisector of CD. Point C is a point on circle with center A. Point H_1 is the intersection of p and ray AC.
Directions: Fill in the blanks below: Since the radius of any circle never changes, it is said to be __________________. This implies radius AC is _____________________. This also means (AH_1 – CH_1) is _________________. Since H_1 lies on p (the perpendicular bisector of DC) , we know ____ = ______. Why is this? Since (AH_1 – CH_1) is ______________, and since CH_1 = ______, it also must be true that the quantity AH_1 – _____ is CONSTANT as well, regardless of where point H_1 lies. This implies that point H_1 is guaranteed to lie on one branch of a/an _______________________with points A and D serving as its __________!