Can the distances from a certain point on the plane to the vertices of a certain square be equal to $1, 4, 7$, and $8$ ?

It is known that in a right triangle: a) The height drawn from the top of the right angle is the geometric mean of the projections of the legs on the hypotenuse; b) the leg is the geometric mean of the hypotenuse and the projection of this leg to the hypotenuse. Are the converse statements true? Formulate them and justify the answer. Is it possible to formulate the criterion of a right triangle based on these statements? If possible, then how? If not, why?

Two circles intersect at points $A$ and $B$. Through point $B$, a straight line intersects the circles at points $C$ and $D$, and then tangents to the circles are drawn through points $C$ and $D$. Prove that the points $A, D, C$ and $P$ - the intersection point of the tangents - lie on the same circle.

Given a cube $ABCDA_1B_1C_1D_1$ with edge $5$. On the edge $BB_1$ of the cube , point $K$ such thath $BK=4$. a) Construct a cube section with the plane $a$ passing through the points $K$ and $C_1$ parallel to the diagonal $BD_1$. b) Find the angle between the plane $a$ and the plane $BB_1C_1$.