臺灣國際科展

平分拋物線.

科展類別
臺灣國際科展
屆次
2008年
科別
數學科
學校名稱
國立臺中女子高級中學
指導老師
賴信志
作者
張?云、楊士潔

摘要或動機

這個研究起源於一個平分圓的問題:在平面上2n +1個點(n∈N),其中任三點不共線,任四點不共圓,任取三點可以畫出唯一的圓,若一半的點在圓內,一半的點在圓外,則此圓為平分圓,Federico
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?

我們的研究題目是:平面上2n +1個在正常位置上的點(n∈N),平分拋物線的個數為何?<說明:定義在平面上2n +1個點(n∈N),其中任三點不共線,任四點不共拋物線,現在我們將對稱軸方向固定時,任兩點連線也不與對稱軸平行,則任取三點可以決定出一個唯一拋物線,若有一半的點在拋物線內,一半的點在拋物線外,則此拋物線稱為平分拋物線。>

我們將研究的主要結果分述如下:

一、證明在平面上2n +1個點(n∈N),平分拋物線個數為定值。


二、證明在平面上2n +1個點(n∈N),平分拋物線個數為n2個。


接著推廣至:若平分拋物線改成(a ∨ b)拋物線,則個數為何?

<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。>

我們將研究的主要結果分述如下:

一、證明在平面上2n +1個點(n∈N),(a ∨ b)拋物線個數為定值。


二、證明在平面上2n +1個點(n∈N),(a ∨ b)拋物線個數為2(ab + a + b +1)個。

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) .



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