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In geometry, the problem of dividing a circle into areas by means of an inscribed polygon with ''n'' sides, in such a way as to ''maximise'' the number of areas created by the edges and diagonals, has a solution by an inductive method. == Lemma == If we already have ''n'' points on the circle and add one more point, we draw ''n'' lines from the new point to previously existing points. Two cases are possible. In the first case (a), the new line passes through a point where two or more old lines (between previously existing points) cross. In the second case (b), the new line crosses each of the old lines in a different point. It will be useful to know the following fact. Lemma. We can choose the new point ''A'' so that case ''b'' occurs for each of the new lines. Proof. Notice that, for the case ''a'', three points must be on one line: the new point ''A'', the old point ''O'' to which we draw the line, and the point ''I'' where two of the old lines intersect. Notice that there are ''n'' old points ''O'', and hence finitely many points ''I'' where two of the old lines intersect. For each ''O'' and ''I'', the line ''OI'' crosses the circle in one point other than ''O''. Since the circle has infinitely many points, it has a point ''A'' which will be on none of the lines ''OI''. Then, for this point ''A'' and all of the old points ''O'', case ''b'' will be true. This lemma means that, if there are ''k'' lines crossing ''AO'', then each of them crosses ''AO'' at a different point and ''k+1'' new areas are created by the line ''AO''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Dividing a circle into areas」の詳細全文を読む スポンサード リンク
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