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 1 of 7: Fix a candidate point

M

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In plain words

Everything about $ABCD$ and $P$ is fixed before $E,F$ are even chosen, so any point built only from them is automatically independent of $E,F$ .

M=(\triangle APD)\cap(\triangle BPC)\setminus\{P\}

Detailed analysis

Let $M$ be the second intersection point (other than $P$ ) of the circumcircles of triangles $APD$ and $BPC$ — the Miquel point of the complete quadrilateral formed by lines $AD,BC,AC,BD$ . Since $A,B,C,D,P$ do not depend on the choice of $E,F$ , neither does $M$ : it is a single fixed point.