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Monday, May 4, 2015

Excircle and circumcircle

In $\triangle ABC$, the $L_1$ be the excircle with respect to $AB$. It touches $CA$ and $CB$ and $M$ and $N$. The circle centered at $C$ and passing through $M$ and $N$ intersects $AB$'s extension at $P,Q$. Show that $L_2$ the circumcircle of $\triangle CPQ$ is tangent to $L_1$, and that the point of tangency between $L_1$ and $L_2$, the point of tangency between $L_1$ and $\triangle ABC$, and $C$ all lie on one line.

Solution Let $X$ be the tangency point of $L_1$ and $AB$. Let $CX$ meets $L_1$ in $Y_1$ and $CX$ meets $L_2$ in $Y_2$. We shall establish that $Y_1 = Y_2$.

Let $C'$ be the diameter of $L_2$ and let $CC'$ meets $AB$ (or its extension) at $D$. Because $CP = CQ$ then it's easy to see that $CD \perp PQ$. Let $s = CM = CN = CP = CQ$.

By power point on $L_1$: $$CX.CY_1 = CM^2 = s^2$$.

By similarity of $\triangle CXD \sim \triangle CC'Y_2$ and $\triangle CC'P \sim \triangle CDP$ : $$CX.CY_2 = CD.CC' = CP^2 = s^2$$

Thus $CY_1 = CY_2$ which means $Y_1 = Y_2$. Now we know that $L_1$ meets $L_2$ in at least one point $Y = Y_1 = Y_2$. Suppose they meet in another point $Z \neq Y$

Let $CZ$ meets $L_1$ in $CZ$ and $X_1 \neq X$. By power point we again have:$CX_1.CZ = s^2$.

Let $CZ$ meets $AB$ in $X_2$. Similar as above, because $Z \in L_2$, we can repeat the similarity argument to establish: $CX_2.CZ = s^2$. This means that $CX_1 = CX_2$, thus $X_1 = X_2 \in L_1, AB$ but that is a contradiction because $L_1$ is only tangent to $AB$, and can only meet $AB$ at $X$.

Now that we've established the tangency between $L_1$ and $L_2$, it's clear that $C,X,Y_1/Y_2$ all lie on one line from the proof.

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