- 10 Nov '07 18:29

I think it should be the exact center (or the center of an inner-circle) of this 25-gon but I don't yet know how to prove this.*Originally posted by Dejection***Autumn 1998 (A Level) Tournament of the Towns**

4. All diagonals of a regular 25-gon are drawn. Prove that no 0 of the diagonals pass through one interior point of the 25-gon. - 10 Nov '07 21:10 / 4 edits

a diagonal is from one vertex to another(an exception is adjacent vertices) which means the if we labled the vertices A,B,C...etc then a diagonal could be from Vertex A to Vertex C and that certainly does not go through the center point.*Originally posted by kbaumen***I think it should be the exact center (or the center of an inner-circle) of this 25-gon but I don't yet know how to prove this.**

Edit: you are probably thinking about the lines of symmetry which of course have to go through the center point. and there are many interior points that the two(or more) diagonals pass through. - 10 Nov '07 22:29

Inscribe the 25-gon in a circle. The only line that could pass through the centre (if that is what you mean), must have a length of d as each vertex in a regular polygon touches the inscribing circle and with be perpendicular to the tangent at the intersection of the vertex and the circle. However, each diagonal must be less than d because the longest diagonals are not perpendicular to the tangent line (this is obvious by inspection).*Originally posted by Dejection***Autumn 1998 (A Level) Tournament of the Towns**

4. All diagonals of a regular 25-gon are drawn. Prove that no 0 of the diagonals pass through one interior point of the 25-gon.

Actually, the whole thing is pretty obvious by inspection.