三角形的内切锥线、和两边相切的锥线交点

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三角形 $A B C$，圆锥曲线 $\mathcal{G}$ 内切于 $A B C$ 且圆锥曲线 $\mathcal{K}$ 与直线 $AB, A C$ 在点 $B, C$ 相切。设圆锥曲线 $\mathcal{G}$ 和 $\mathcal{K}$ 相交于两点 $P, Q$。证明从点 $P, Q$ 到圆锥曲线 $\mathcal{G}$ 的切线的交点在圆锥曲线 $\mathcal{K}$ 上。
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clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);

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On theorem generators in plane geometry

Problem 3. Let given triangle $A B C$, let conic $\mathcal{G}$ is inscribed in $A B C$ and conic $\mathcal{K}$ tangent to lines $A B, A C$ at points $B, C$. Let conics $\mathcal{G}$ and $\mathcal{K}$ intersects at two points $P, Q$. We prove that tangents to conic $\mathcal{G}$ from points $P, Q$ intersects on conic $\mathcal{K}$.

Proof. Consider algebraical group $G:=\operatorname{PGL}\left(\mathbb{C}^2\right)$ as group of projective transformations of plane $\mathbb{C}^2$. Consider object $\mathcal{X}$ as sets of conic $\mathcal{K}$ and lines $l, x, y$ which are tangent to it and where $x \| y, x$ is perpendicular to $l$. Consider morphism $\pi: \mathcal{X} \times G \rightarrow$ $\mathcal{Y}$, where $\mathcal{Y}$ is sets : conic and three lines which are tangent to it, $\pi((\mathcal{K}, l, x, y), g):=$ $(g(\mathcal{K}), g(l), g(x), g(y)) \in \mathcal{Y}$. Easy to check that $\pi$ is dominating. If for any set $\left(\mathcal{K}, l_1, l_2, l_3\right) \in$ $\mathcal{Y}$ we construct conic $\mathcal{C}$ which is tangent to lines $l_2, l_3$ at points $l_1 \cap l_2, l_1 \cap l_3$, and consider intersection point of tangents from points $P, Q:=\mathcal{K} \cap \mathcal{C}$ to conic $\mathcal{K}$, and also consider conic $\mathcal{C}$, then we get morphism $\psi: \mathcal{Y} \rightarrow \mathcal{Z}$, where $\mathcal{Z}$ is sets : point on plane and conic. Easy to check that if $\phi:=\pi \circ(\psi, e)$, then next diagram is commutative:\begin{xy}
\xymatrix {
&\mathcal X \times G \ar[dl]_\pi \ar[dr]^\phi\\
\mathcal Y \ar[rr]^\psi &&\mathcal Z
}
\end{xy}From density of $π$ we get that $\operatorname{Im}ψ ⊆ \operatorname{Im}φ$ and from problem 2 we know that $\operatorname{Im}(φ)$ is conics with points on them.

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2# 中的证明很难理解！有人能提出更简单的证明吗？

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1 楼的图无法再加载, 什么原因造成的