Problem 5

Let

ABCD

be a fixed convex quadrilateral with

BC = DA

and

BC \nparallel DA

. Let two variable points

E

and

F

lie on the sides

BC

and

DA

, respectively, and satisfy

BE = DF

. The lines

AC

and

BD

meet at

P

, the lines

BD

and

EF

meet at

Q

, the lines

EF

and

AC

meet at

R

. Prove that the circumcircles of the triangles

PQR

, as

E

and

F

vary, have a common point other than

P

Step 3 of 7: The same rotation also sends

E

F

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In plain words

A rotation sending one segment onto another sends each point to the point at the matching fractional distance; equal fractions $BE/BC=DF/DA$ make $E$ and $F$ a matching pair.

\frac{BE}{BC}=\frac{DF}{DA}\ \Longrightarrow\ \text{the same rotation sends}\ E\mapsto F

Detailed analysis

The rotation of the previous step sends the point of $BC$ at fractional distance $BE/BC$ from $B$ to the point of $DA$ at the same fractional distance from $D$ . Since $BE=DF$ and $BC=DA$ , that fraction is $BE/BC=DF/DA$ , and the image point is exactly $F$ . So this single fixed rotation about $M$ sends $B\mapsto D$ , $C\mapsto A$ , and $E\mapsto F$ simultaneously.