We provide a theory as to why the locus of a triangle center over Poncelet 3-periodics in an ellipse pair is an ellipse or not. For the confocal pair (elliptic billiard), we show that if the center can be expressed as a fixed affine combination of barycenter, circumcenter, and mittenpunkt (which is stationary over the confocal family), then its locus will be an ellipse. We also provide conditions under which a particular locus will be a circle or a segment. We also analyze locus turning number and monotonicity with respect to vertices of the 3-periodic family. Finally we write out expressions for the convex quartic locus of the incenter for a generic Poncelet family, conjecturing it can only be an ellipse if the pair is confocal.
翻译：我们给出了一个理论,说明为什么庞塞莱特三期半圆形三角中心在椭圆对的三角中心位置是椭圆的或不是椭圆的。对于圆锥对,我们展示了如果中心可以表现成一个固定的方形组合,由甘蓝、环球和Mittenpunkt(固定在圆形家族之上)组成,那么其中心就是椭圆。我们还提供了一个特定方圆或一个区段的条件。我们还分析了三周期家庭脊椎的圆形转转转转数和单向性。最后,我们为普通庞塞特家族写出中心方圆形圆形圆形圆形圆形圆形的表达方式,当对子是圆形时,它只能是一个椭圆形。
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