In the triangle ABC with circumcentre O ,AB = AC, D is the midpoint of AB,
and E is the centroid of triangle ACD. Prove that OE is perpendicular to CD.
I don't think there is sufficient information to solve this, are there no numbers for midpoints and sides given? the only way to prove this would be to take the slope of both sides and if they are negative reciprocals, they are perpendicular, which we cannot do with the information provided.
I can't offer you a proof but a more or less accurate construction.
1. The dotted circle is necessary to get the sides AB and AC with equal length. The green circle is the circumcircle of ABC.
2. The angle of 90° is not constructed but measured.
3. If I change the shape of the triangle, which is quite easy with the program I use, the measured angle stay constant at 90°.