# 平分拋物線.

2008年

## 摘要或動機

Ardila 教授在America Monthly 111 期[2]中發表了一篇論文，證明平分圓的個數為n2個。我們研究的目的是：如果將圓改成拋物線，則平分拋物線的個數是否為一定值?
若為定值，則為多少個?

<說明：若a 個點在拋物線內，b 個在拋物線外，或b 個點在拋物線內， a 個點在拋物線外 (a + b = 2n − 2)，則稱此拋物線為(a ∨ b)拋物線。>

This study originated from a question of “The Number of Halving Circles＂: Setting
2n +1 points in the plane is in general position if no three of the points are collinear
and no four are concyclic. We call a circle halving with respect to those 2n +1
points if it has three points of those 2n +1 points on its circumference, n −1 points
in its interior, and n −1 in its exterior. Then we call this circle “Halving Circle.＂
Professor Federico Ardila issued a paper in the America Monthly 111 [2]. The goal
of that paper is to prove the following fact: any set of 2n +1 points in general
position in the plane has exactly n2 halving circles. The purpose we
make the study of is: If we turn circles into parabolas, how many Halving Parabolas
are there?

The title we make the study of is: Setting 2n +1 points in the plane (n∈N) , how
many Halving Parabolas are there?

<Interpretation: Setting 2n +1 points in the plane is in general position if
no two of the points are parallel with the fixing direction of the axis of symmetry,
no three of the points are collinear and no four are on the same parabola. We call
a parabola halving with respect to those 2n +1 points if it has three points of
those 2n +1 points on its curve, n −1 points in its interior, and n −1 in its exterior.
Then we call this circle “Halving Parabola.＂>

We show our main effect below:

1. Proving that 2n +1 points in the plane (n∈N) , the number of Halving Parabolas
is constant.

2. Proving that 2n +1 points in the plane (n∈N) , the number of Halving Parabolas
is n2 .

Spread: If we turn Halving Parabolas into (a ∨ b) Parabolas, how many (a ∨ b) Parabolas
are there?

<Interpretation: Setting 2n +1 points in the plane is in general position if
no two of the points are parallel with the fixing direction of the axis of symmetry,
no three of the points are collinear and no four are on the same parabola. We call
a parabola separating with respect to those 2n +1 points if it has three points
of those 2n +1 points on its curve, a points in its interior and b in its exterior,
or b points in its interior and a in its exterior. Then we call this circle “(a
∨ b) Parabola.＂>

We show our main effect below:

1. Proving that 2n +1 points in the plane (n∈N) , the number of (a ∨ b) Parabolas
is constant.

2. Proving that 2n +1 points in the plane (n∈N) , the number of (a ∨ b) Parabolas
is 2(ab + a + b +1) .

「為配合國家發展委員會「推動ODF-CNS15251為政府為文件標準格式實施計畫」，以及 提供使用者有文書軟體選擇的權利，本館檔案下載部分文件將公布ODF開放文件格式， 免費開源軟體可至LibreOffice 下載安裝使用，或依貴慣用的軟體開啟文件。」