Question 3
A circle $(L)$ with center $O$ is inscribed in a rhombus $ABCD$. A variable tangent to $(L)$ meets the lines $AB,AD,BC,CD$ at $M,N,P,Q$, respectively.
(a) Conjecture a relation between the lengths $BM$ and $DN$, and prove it. What other noteworthy relations occur in the figure? Discover as many as possible.
(b) Four circles, all passing through $O$ and centered respectively at $A,B,C,D$, intersect the four segments $AB,BC,CD,DA$ at eight points. Examine the special properties of the convex octagon having these points as vertices, and prove them. Apply this to constructing a regular octagon.
(c) Formulate a solid-geometry problem by rotating the figure of the preceding problem about the axis $AC$.
(a) Conjecture a relation between the lengths $BM$ and $DN$, and prove it. What other noteworthy relations occur in the figure? Discover as many as possible.
(b) Four circles, all passing through $O$ and centered respectively at $A,B,C,D$, intersect the four segments $AB,BC,CD,DA$ at eight points. Examine the special properties of the convex octagon having these points as vertices, and prove them. Apply this to constructing a regular octagon.
(c) Formulate a solid-geometry problem by rotating the figure of the preceding problem about the axis $AC$.