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 5 of 7:

M

lies on the circle through

Q,D,F

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In plain words

The same trick works with the other pair of corresponding points, $B\mapsto D$ and $E\mapsto F$ , producing a second circle through $M$ .

Q=BD\cap EF\ \Longrightarrow\ M\in(\triangle QDF)

Detailed analysis

The very same rotation about $M$ also sends $B\mapsto D$ and $E\mapsto F$ . Applying the spiral-similarity lemma to this pair places $M$ on the circumcircle of the triangle formed by $Q=BD\cap EF$ together with $D$ and $F$ , i.e. $M\in(\triangle QDF)$ .