Let $ABC$ an isosceles triangle and $\Delta$ a variable line passing trough $A$. We denote $C’$ the image of $C$ through the reflection across $\Delta$ axis.
1) Find the locus of $C’$ when $\Delta$ varies.
This question is easy to solve since $AC’=AC$ the locus of $C’$ is the circle centered at $A$ and passing through $C$.
2) Here’s the problem: Let $M=(BC’)\cap(\Delta)$. Find the locus of $M$.
Here I conjecture that this locus is the circumcircle of $ABC$ but I fail to prove it. Here’s a picture.
Hint: Observe that due to symmetry $AB=AC=AC'$ and hence $B,C,C'$ lie on a circle centered at $A$. Thus $\angle CAB=2\cdot \angle CC'B=\angle CMB...$