https://file.aiquickdraw.com/mathbot/file/1732028846835z68d5qcj.jpg

Let's break the problem into parts:

### 1°: Show that the triangle \( \triangle MON \) is right at \( O \).

**Steps:**
1. **Geometry setup:**
   - \( AB \) is the diameter of the circle with center \( O \).
   - By the property of a circle, \( AB \) subtends a right angle at any point \( P \) on the circle not on \( AB \). Thus, \( \angle APB = 90^\circ \).

2. **Tangent properties:**
   - Tangents at \( A \), \( B \), and \( P \) form intersections at \( M \) and \( N \).
   - Since \( OM \) and \( ON \) are radii drawn to the tangents at \( A \) and \( B \), they are perpendicular to the tangents. Therefore, \( OM \perp MA \) and \( ON \perp NB \).

3. **Triangle \( \triangle MON \):**
   - \( OM \) and \( ON \) are radii, and \( MN \) is the chord of the circle.
   - The points \( M \), \( O \), and \( N \) form a right triangle at \( O \) because \( OM \perp ON \) (as both are perpendicular to their respective tangents).

### 2°: Show that the circle circumscribed about \( \triangle MON \) is tangent to the line \( AB \).

**Steps:**
1. **Circumcircle of \( \triangle MON \):**
   - The circumcircle of \( \triangle MON \) is centered at \( O \) because \( O \) is the point of intersection of the perpendicular bisectors of \( MN \).
   - The radius of this circumcircle is \( OM = ON \) (equal radii from \( O \)).

2. **Tangency to \( AB \):**
   - Line \( AB \) is the diameter of the circle.
   - The tangents at \( A \) and \( B \) are perpendicular to the radius. Since \( M \) and \( N \) lie on these tangents, the circumcircle of \( \triangle MON \) is tangent to the diameter \( AB \) due to the tangential property of radii to chords.

Let me know if you’d like further clarification, diagrams, or expanded reasoning.

### Relative Questions:
1. How can the Pythagorean theorem be used to verify that \( \triangle MON \) is right at \( O \)?
2. What is the relationship between the radius of the circumcircle of \( \triangle MON \) and the diameter of the main circle \( C \)?
3. Why is the tangency property essential in proving the second part of the problem?
4. How does the tangent at \( P \) interact geometrically with points \( M \) and \( N \)?
5. What other properties of the circle \( C \) can help reinforce the conclusions about \( \triangle MON \)?

### Tip:
When working with circles, always focus on relationships involving radii, diameters, and tangent properties to uncover key geometric insights.

[AB] is a diameter of a circle (C) of center O. P is a point of (C) distinct from A and B. The tangent at P to (C) cuts the tangents at A and B in M and N respectively.
1° Show that the triangle MON is right at O.
2° Show that the circle circumscribed about triangle MON is tangent to the line (AB).

Explore a geometry problem involving a circle's diameter, tangents, and circumcircle properties. Learn how to prove triangle MON is a right triangle and that its circumcircle is tangent to the line AB, using properties of tangents and circles.

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