To solve the problem described in the image, I will break it into its components and guide you step by step:

Step 1: Draw the figure

1. Draw a circle $U$ with center $J$, and diameter $CD$.
2. Mark a point $H$ on the circle distinct from $C$ and $D$.
3. Draw the tangent at $H$ to the circle $U$. Extend this tangent line such that it cuts the tangents at $C$ and $D$ at points $M$ and $N$, respectively.

Step 2: Show that the triangle $MJN$ is right at $J$

1. The tangent at $H$ is perpendicular to the radius $JH$. Since $H$ lies on the circle, $JH$ is perpendicular to the tangent.
2. Using the properties of tangents from a common point to a circle, $MJ$ and $NJ$ are perpendicular to $CD$.
3. Therefore, $\triangle MJN$ is a right triangle with a right angle at $J$.

Step 3: Prove that $CDNM$ is a right trapezoid

1. The tangents at $C$ and $D$ to the circle are parallel (both are perpendicular to the radius $JC$ and $JD$).
2. Since $MN$ is formed by extending the tangents, $CDNM$ is a trapezoid with parallel sides $CD$ and $MN$.
3. To show it's a right trapezoid, observe that $MJ$ and $NJ$ (which are sides of the trapezoid) are perpendicular to $CD$, forming right angles at $M$ and $N$.

Step 4: Prove that $JL$ is perpendicular to $CD$

1. $L$ is the midpoint of $MN$, the hypotenuse of $\triangle MJN$.
2. By the Midpoint Theorem, the line joining the midpoint of the hypotenuse to the opposite vertex is perpendicular to the base of the triangle.
3. Hence, $JL$, the median from $J$ to $L$, is perpendicular to $CD$.

Step 5: Prove that the circle circumscribed about $MJN$ is tangent to $CD$

1. The circle circumscribed about $\triangle MJN$ has its center at the midpoint of $MN$, which is $L$, and radius equal to half of $MN$.
2. Since $JL$ is perpendicular to $CD$, $JL$ acts as the radius from the center of the circumscribed circle to $CD$.
3. Hence, the circumscribed circle is tangent to $CD$.

If you’d like, I can draw or detail any specific step further. Let me know!

Here are 5 related questions for exploration:

1. How can you mathematically find the equation of the tangents at $C$ and $D$?
2. Why is the tangent perpendicular to the radius of a circle at the point of tangency?
3. Can you prove the Midpoint Theorem geometrically for this case?
4. How does the property of parallel tangents contribute to the trapezoidal shape of $CDNM$?
5. How would you calculate the radius of the circumscribed circle around $\triangle MJN$?

Tip: Always start with a well-labeled diagram to keep the geometry visually intuitive and manageable!