平分拋物線.
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?
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?
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?
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?
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) .