P is the circumcenter of Delta ABC If BP=9x-29,AP=5x-1 and PF=15 find FC.

<br />1. In this problem, we must utilize a property of the circumcenter of a triangle: that the circumcenter is equidistant from all vertices of the triangle. In other words, the radius of the circumscribed circle around triangle ABC is the same length for AP, BP, and CP.<br /><br />2. Given that BP = 9x - 29 and AP = 5x - 1, and since both are radii of the same circumcircle, we can set them equal to each other to solve for x.<br /><br />3. Since PF is part of the diameter, and since FC is part of the diameter, in a circle, the diameter is twice the length of the radius. PF and FC split the diameter into two segments, and PF has been given a length of 15, we must find the value of FC to complete the diameter and ensure we adhere to the diameter's double radius property.<br /><br />4. Thus, our steps are as follows: <br />- Firstly, solve for x by equating BP and AP. <br />- Identify the length of BP which will also serve as the length of the diameter. <br />- Since we know PF, we then subtract PF from the diameter to find FC.<br /><br />5. With the values provided, we are able to do the calculations as such:<br />- Solve for x in the equation 9x - 29 = 5x - 1.<br />- Calculate the length of BP once we have the value of x.<br />- Calculate FC as the diameter minus PF.<br /><br />The thought process above is guided by the rules of geometry and the properties particular to circles and their circumcenters.