# Similar Triangles Determined by Miquel Circles and the Circumcircle

Similar Triangles Determined by Miquel Circles and the Circumcircle

Let ABC be a triangle and let , , and be points on BC, CA, and AB, respectively. Suppose that the circumcircles of , , and (the Miquel circles) intersect the circumcircle of ABC at points , , and , respectively, where ≠A, ≠B, and ≠C. Let , , and be symmetric to , , and with respect to the midpoints of BC, CA, and AB, respectively. Then and are similar.

A

1

B

1

C

1

AB

1

C

1

BC

1

A

1

CA

1

B

1

A

2

B

2

C

2

A

2

B

2

C

2

A

3

B

3

C

3

A

1

B

1

C

1

A

2

B

2

C

2

A

3

B

3

C

3