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 7 of 7:

M

is the common point on every circle

(PQR)

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M\in(\triangle PQR)

Detailed analysis

Since $M$ is the Miquel point of the four lines $DA,EF,BD,AC$ , it lies on the circumcircle of every triangle they determine, in particular on $(\triangle PQR)$ — the triangle cut out by lines $EF,BD,AC$ . Because $M$ is defined solely from the fixed points $A,B,C,D,P$ (Step 1), it is the same point for every valid choice of $E,F$ , and it is generically different from $P$ . This is the required common point.