OME 1966 Problem 5

The length of the hypotenuse $BC$ of a right triangle $ABC$ is $a$ , and on it the points $M$ and $N$ are taken such that $BM = NC = k$ , with $k < a/2$ . Assuming that (only) the data $a$ and $k$ are known, calculate: a) The value of the sum of the squares of the lengths $AM$ and $AN$ . b) The ratio of the areas of triangles $ABC$ and $AMN$ . c) The area enclosed by the circle that passes through the points $A, M' , N'$ , where $M'$ is the orthogonal projection of $M$ onto $AC$ and $N'$ that of $N$ onto $AB$ .