My friend recently gave me a problem that I was interested in, but could not completely solve. I've already constructed a diagram that roughly represents the problem, and it's below.
Consider a trapezoid $ABCD$ with $AB \parallel CD$. Additionally, side $BC = CD = 43$ and $AD \perp BD$. Given that the length between the midpoint of diagonal $BD$ and intersectional of diagonals $AC$ and $BD$ is $11$, find the side length $AD$. Express your answer in simplified radical form.
So I gave some names to the other points, calling the midpoint of the diagonal $M$ and the intersection $I$. The first thing that I noted was that because $BC = CD$, I found that drawing it down to the midpoint of diagonal $BD$ created two right triangles. Additionally, I noted congruent angles within the triangle $CPO$ and triangle $ADO$, so I deduced they were similar. But I'm not sure how to proceed from here, as I'm not able to establish the side lengths of any more segments. Does anyone know how to proceed?