Two Congruent Circles by Reflection: What is this about?
A Mathematical Droodle

Explanation

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The applet attempts to illustrate a problem by Quang Tuan Bui:

I is the incenter and X an arbitrary point in the plane of ΔABC. B_i and B_x are reflections of B in AI and AX, and similarly C_i and C_x. Show that the circumcircles CB_iB_x and BC_iC_x are congruent.

Proof

The product of the reflections in two intersecting axes is a rotation through twice the angle between them, with the center at the point of intersection. So that C_i is obtained from C_x by a rotation around A which also maps B_i onto B_x.

It follows that triangles C_iAC_x and B_iAB_x are both isosceles with the same apex angle which makes their base angle also equal. In particular,

Now, let B_j be the reflection of B_i in AX. We have

implying that the quadrilateral BC_iC_xB_j is cyclic. The circle circumscribed about that quadrilateral is a circumcircle of two triangles, BC_iC_x and BC_xB_j. But ΔBC_xB_j is the reflection of ΔB_xCB_i in AX so that their circumcircles are congruent. This shows that the circumcircles BC_iC_x and CB_iB_x are also congruent.

Naturally, the two circles are the reflections of each other in AX.

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