USAJMO 2026 Question 3

Let $ABC$ be acute and scalene, with no angle $60^\circ$, and circumcircle $\omega$. Let $\Delta_B$ be the equilateral triangle inscribed in $\omega$ with one vertex $B$, and let $\ell_B$ pass through its other two vertices. Define $\Delta_C,\ell_C$ analogously. Let $Y=AC\cap\ell_B$ and $Z=AB\cap\ell_C$. The circumcircle of $AYZ$ meets $\omega$ again at $P\ne A$. Let $D=BC\cap YZ$ and $E=PA\cap YZ$. Prove $PE=PD$.

Proposed by Rogelio Guerrero.

Manual essay; 1 editorial point, not official contest scoring.