<s>[INST] Finding $\frac{DK}{DI}$ In triangle $ABC,$ where $AB = 8, AC = 7,$ and $BC = 10,$ $I$ is the incenter. If $AI$ intersects $BC$ at $K$ and the circumcircle of $\triangle ABC$ at $D,$ find $\frac{DK}{DI}.$

I first drew a diagram, but I was unsure where to go from here.

 [/INST] I have doubts that you correctly compute the result, since the value $IK$ given in comments is incorrect. The correct result is:
$$\frac{DK}{DI}=\frac23.
$$
The details are given below.

Let $x,y,z$ being the distances from the vertices $A,B,C$ to the tangent points of the incircle. From the equations $x+y=c, y+z=a, z+x=b$ one obtains $x=\frac{b+c-a}2=s-a$, where $s$ is the semiperimeter. Then:
$$AI=\sqrt{x^2+r^2}=\sqrt{(s-a)^2+\frac{(s-a)(s-b)(s-c)}s}=\sqrt {\frac {bc (s-a)}{s}}=2\sqrt{\frac{14}5}.$$
The angle bisector length is:
$$AK=\sqrt{bc\left[1-\left(\frac  a{b+c}\right)^2\right]}=\frac {\sqrt{4bc (s-a)s}}{b+c}=\frac{10}3\sqrt{\frac{14}5}.$$
By the power of point $K$ and angle bisector theorem:$$\color {red}{DK}=\frac{BK\cdot KC}{AK}=\frac{a^2\frac{bc}{(b+c)^2}}{AK}=\frac{a^2}{b+c}  \sqrt {\frac{bc}{4 (s-a)s}}=\frac83\sqrt{\frac{14}5},$$
so that:
$$\color {red}{DI}=DK+AK-AI=4\sqrt{\frac{14}5}.$$

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