Triangle $ABC$ has side lengths $AB=7, BC=8,$ and $CA=9.$ Its incircle $\Gamma$ meets sides $BC,CA,$ and $AB$ at $D,E,F$ respectively. Let $AD$ intersect $\Gamma$ at a point $P \neq D.$ The circle passing through $A$ and $P$ tangent to $\Gamma$ intersects the circle passing through $A$ and $D$ tangent to $\Gamma$ at a point $K \neq A.$ Find $\tfrac{KF}{KE}.$
I was not sure how to approach this problem and am still not completely sure. First, I drew an almost to scale diagram (it wasn't 100% to scale because I was not sure how to perfectly construct the circle passing through $A$ and $D$ tangent to $\Gamma.$ (It would be helpful for a strategy to precisely construct the circle passing through $A$ and $D$ tangent to $\Gamma$ though). The diagram is shown below
By some approximation, the ratio was around $37/42$ but I'm not sure if this is correct, especially because it is an unrigorous approximation. I have an idea to set coordinates and bash out the coordinates for $K,F,E$ but this would be tedious and an unclean strategy. May I have some help? Thanks in advance.
Best Answer
Let the circle through $A$ and $D$ tangent to $BC$ be $\omega_1$ and the circle through $A$ and $P$ tangent to $\Gamma$ be $\omega_2$. Then by radical axes on $\omega_1$, $\omega_2$ and $\Gamma$, we see that $AK$, $BC$, and the common tangent of $\omega_2$ and $\Gamma$ are concurrent at a point, say $X$. Now since $PEDF$ is a harmonic quadrilateral, we see that $EF$ passes through $X$, so $XE \cdot XF = XD^2 = XK \cdot XA$, and consequently $AKFE$ is cyclic. Now let $\omega_1$ intersect $AC$ and $AB$ at $Q$ and $R$ respectively. Then by spiral similarity, $\triangle KEQ \sim \triangle KFR$, so $\frac{KF}{KE} = \frac{FR}{EQ}$, and all there is left is to compute these two lengths. The side lengths of the triangle are $a = 8$, $b = 9$ and $c = 7$, and the semiperimeter $s = 12$. Since $D$, $E$, $F$ are the touchpoints of the incircle with the sides, we have $BD = BF = s - b = 12 - 9 = 3$, and $CD = CE = s - c = 12 - 7 = 5$. Therefore, $$ EQ = CE - CQ = CD - \frac{CD^2}{CA} = 5 - \frac{25}{9} = \frac{20}{9} $$ and similarly, $$ FR = BF - BR = BD - \frac{BD^2}{BA} = 3 - \frac{9}{7} = \frac{12}{7}. $$ Hence finally, $$ \frac{KF}{KE} = \frac{FR}{EQ} = \frac{12}{7} \cdot \frac{9}{20} = \frac{27}{35}. $$
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