Thứ Sáu, 12 tháng 9, 2014

Let $A,B,C,D$ are concyclic; $B,N,P,C$ are concyclic; $M,A,D,Q$ are concyclic. Let $A,B,C,D,M,N,P,Q$ are on conic. Prove that: $M,N,P,Q$ are concyclic.

Solution by: Luis González

$\mathcal{C}$ the given conic with axes $\ell_1,\ell_2.$ Arbitrary circle $\omega$ through $A,D$ cuts $\mathcal{C}$ again at $X,Y.$ $U,V$ are points on $\mathcal{C},$ such that the tangents of $\mathcal{C}$ at $U,V$ are parallel to $XY,AD,$ respectively. Let $I$ be the intersection of these tangents and $J \equiv AD \cap XY.$ By generalized power of point, we have

$\frac{IV^2}{IU^2}=\frac{JA \cdot JD}{JX \cdot JY}=1 \Longrightarrow IU=IV.$

Hence, $I$ is either on $\ell_1$ or $\ell_2$ $\Longrightarrow$ $AD$ and $XY$ are equal inclined to $\ell_1,\ell_2.$ Thus when $\omega$ varies, keeping $A,D$ fixed, all lines $XY$ go through a fixed direction. As a result, we deduce that $AD \parallel NP$ and $BC \parallel MQ.$ If $\odot(MNP)$ cuts $\mathcal{C}$ again at $Q^*,$ then $BC \parallel MQ^*$ $\Longrightarrow$ $Q \equiv Q^*$ $\Longrightarrow$ $M,N,P,Q$ are concyclic.

Dao's blog

Thứ Sáu, 12 tháng 9, 2014

35-Three circles and a conic

Let $A,B,C,D$ are concyclic; $B,N,P,C$ are concyclic; $M,A,D,Q$ are concyclic. Let $A,B,C,D,M,N,P,Q$ are on conic. Prove that: $M,N,P,Q$ are concyclic.

Không có nhận xét nào:

Lưu trữ Blog

Blog bạn bè

Thứ Sáu, 12 tháng 9, 2014

35-Three circles and a conic

Let are concyclic; are concyclic; are concyclic. Let are on conic. Prove that: are concyclic.

Không có nhận xét nào:

Let $A,B,C,D$ are concyclic; $B,N,P,C$ are concyclic; $M,A,D,Q$ are concyclic. Let $A,B,C,D,M,N,P,Q$ are on conic. Prove that: $M,N,P,Q$ are concyclic.