[Theorem of the tangent and secant; the product of the entire secant and its external part [ Auxiliary similar triangles

In trapezoid $A B C D$, the lateral side $A B$ is perpendicular to the base $B C$. A circle passes through points $C$ and $D$ and is tangent to line $A B$ at point $E$.

Find the distance from point $E$ to line $C D$, if $A D=4, B C=3$.

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Apply the tangent-secant theorem.

## Solution

Let $T$ be the intersection point of lines $A B$ and $C D$, $P$ be the projection of point $E$ onto line $C D$, and $Q$ be the projection of point $C$ onto line $A D$.

Then $Q D=A D-A Q=A D-B C=1$.

From the similarity of right triangles $T B C$ and $T A D$, we find that $T C=3 C D$. Therefore, $T E^{2}=T D \cdot T C=12 C D^{2}$. From the similarity of right triangles $T P E$ and $C Q D$, we get $E P=\frac{T E \cdot Q D}{C D}=\frac{2 C D \sqrt{3}}{C D}=2 \sqrt{3}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_df2bbd76ac4dd2f0f20fg-15.jpg?height=420&width=712&top_left_y=1628&top_left_x=679)

## Answer

$2 \sqrt{3}$.