Difficult Geometry - Circle #1

Geometry Level pending

Circles \(O_1\) and \(O_2\) intersect with each other at points \(\text{A}\) and \(\text{B}\).

\(O_1\) and \(O_2\) has a radius of \(3\) and \(5\), and has a center of \({\text{C}}_1\) and \({\text{C}}_2\), respectively.

Point \(\text{D}\) internally divides \(\overline{\text{AB}}\) with a ratio of \(1:3\).

\(\overline{\text{EF}}\) and \(\overline{\text{KP}}\) intersect at point \(\text{D}\).

\(\overline{\text{EF}}\) is a chord of \(O_1\), with point \(\text{F}\) being inside circle \(O_1\) and \(\text{E}\) being outside it, satisfying \(\overline{\text{AE}}<\overline{\text{BE}}\) and \(\overline{\text{AF}}>\overline{\text{BF}}\).

Point \(\text{K}\) is on circle \(O_2\) and inside circle \(O_1\). Also, it satisfies \(\angle\text{EFK}=20^{\circ}\), \(\angle\text{EKF}=130^{\circ}\) and \(\angle{\text{C}}_1\text{FK}<\angle{\text{C}}_1\text{FE}\).

Point \(\text{P}\) satisfies \(\angle\text{EPF}=50^{\circ}\).

Find the value of \(\overline{{\text{C}}_2 \text{P}}\).

If you think the given information is too less to figure out the answer, submit \(-1\) as your answer. And if you think the given information doesn't make sense, submit \(0\) as your answer.


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