2017-TST-9 By Yijun Yao

by EthanWYX2009, Jan 30, 2025, 4:47 AM

Problem.Let $ABCD$ be a quadrilateral and let $l$ be a line. Let $l$ intersect the lines $AB,CD,BC,DA,AC,BD$ at points $X,X',Y,Y',Z,Z'$ respectively. Given that these six points on $l$ are in the order $X,Y,Z,X',Y',Z'$, show that the circles with diameter $XX',YY',ZZ'$ are coaxal.
Proof. Consider $\ell :y=c$ where $c$ is moving. $X(u,c),Y(v,c),Z(w,c),X'(u',c),Y'(v',c),Z'(w',c).$
The Analytical formula of $\odot (XX')$ is $$(x-(u+u')/2)^2+(y-c)^2=(u-u')^2/4\iff x^2-(u+u')x+uu'+(y-c)^2=0.$$To prove the three functions are linearly dependent, we only need
$$\frac{u+u'-v-v'}{uu'-vv'}=\frac{u+u'-w-w'}{uu'-ww'}.$$However we may write $u=P(c)$ where $\deg P=1,$ thus the above equality is $Q(c)$ with degree 3. Therefore we only need to plug in 4 special $c,$ and using $A,B,C,D$ we are done! $\Box$

I will post no geometry in my main blog, this is where we can BASH!!!

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2017-TST-9 By Yijun Yao
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