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The pentagram revisited

Source: problem 14 (G4) of QEDMO 1; created by myself

November 7, 2005

trigonometrygeometry proposedgeometry

Problem Statement

In the following, the abbreviation

g \cap h

will mean the point of intersection of two lines

g

and

h

. Let

ABCDE

be a convex pentagon. Let

A^{\prime}=BD\cap CE

,

B^{\prime}=CE\cap DA

,

C^{\prime}=DA\cap EB

,

D^{\prime}=EB\cap AC

and

E^{\prime}=AC\cap BD

. Furthermore, let

A^{\prime\prime}=AA^{\prime}\cap EB

,

B^{\prime\prime}=BB^{\prime}\cap AC

,

C^{\prime\prime}=CC^{\prime}\cap BD

,

D^{\prime\prime}=DD^{\prime}\cap CE

and

E^{\prime\prime}=EE^{\prime}\cap DA

. Prove that:

\frac{EA^{\prime\prime}}{A^{\prime\prime}B}\cdot\frac{AB^{\prime\prime}}{B^{\prime\prime}C}\cdot\frac{BC^{\prime\prime}}{C^{\prime\prime}D}\cdot\frac{CD^{\prime\prime}}{D^{\prime\prime}E}\cdot\frac{DE^{\prime\prime}}{E^{\prime\prime}A}=1.

Darij

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