Circles
Introduction
The objective of this task is to explore the relationship between the positions of points within circles that intersect. The first figure illustrates circle C1 with radius r, centre O, and any point P. r is the distance between the centre O and any point (such as A) of circle C1.

Figure 1 The second diagram shows circle C2 with radius OP and centre P, as well as circle C3 with radius r and centre A. An intersection between C1 and C2 is marked by point A. The intersection of C3 with OP is marked by point P’. Figure 2

Through this investigation I will examine how the r values correlate with the values of OP in determining the length of OP’ when r is held as a constant variable and the value of OP is the variable that is subject to change. I will then venture on to study the inverse, the relationship when the r values becomes the variable that is changed and the OP value is held constant. r as a Constant

If we let the value of r be equal to 1, we can use that information to find the length of OP’ when OP=2, 3, and 4. The first thing one can deduce is that by using the points A, O, P’, and P two isosceles triangles can be formed; ∆AOP and ∆AOP’. To rationalize this assertion through an analytic approach it should first be understood that all line segments that connect the center of a circle to its perimeter, or circumference, is considered to be its radius. Because P’ and O are both located within the circumference of C3 and connect to its center, A, we can establish that AP’ and OA are radii of C3. Likewise, points O and A are situated within the circumference of C2 and connect to its center, P, meaning that both OP and AP can be regarded as the radius of C2. By definition all the radii within a same circle are of equal length, thus proving that OA and AP’ are of the...

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