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