Homework is given to a child who is not ready for a math competition - (4).

$(A, m)$ and $(B, n)$ are fixed circles (and $m > n > AB = p$ is fixed) that intersect at points $C$ and $D$. A line passes through $D$ that intersects $(A)$ and $(B)$ respectively at $E$ and $F$. $DG \perp CE$ at $G$, $DH \perp CF$ at $H$. $GA$ and $HB$ intersect at $I$.

a. Determine the position of $EF$ such that $CI$ has the shortest possible length.

b. Determine the locus of $I$ as $EF$ moves but always passes through $D$ .

#Geometry

Easy Math Editor

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$