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Area apef/area abp does not depend on p

Source:

October 30, 2005
geometrycircumcircle

Problem Statement

Let PP be a point inside a triangle ABCABC such that ∠PBC=∠PCA<∠PAB\angle PBC = \angle PCA < \angle PAB. The line PBPB meets the circumcircle of triangle ABCABC at a point EE (apart from BB). The line CECE meets the circumcircle of triangle APEAPE at a point FF (apart from EE). Show that the ratio ∣APEF∣∣ABP∣\frac{\left|APEF\right|}{\left|ABP\right|} does not depend on the point PP, where the notation ∣P1P2...Pn∣\left|P_1P_2...P_n\right| stands for the area of an arbitrary polygon P1P2...PnP_1P_2...P_n.
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