Problem
GEO-B2-M10-P012 Four Circles from Four Lines
#12
★★★☆☆ Level 3 of 5
In triangle \(ABC\), points \(D\in AB\), \(E\in AC\), and lines \(BE\) and \(CD\) meet at \(P\). Prove that circles \((ABE)\), \((ACD)\), \((BDP)\), \((CEP)\) have one common point.
Consider the complete quadrilateral formed by lines \(AB,AC,BE,CD\).
The four lines \(AB,AC,BE,CD\) form a complete quadrilateral. Its four circles are exactly \((ABE)\), \((ACD)\), \((BDP)\), \((CEP)\). By Miquel's theorem, they pass through one point.
This is a problem on recognising a complete quadrilateral inside a triangle.