1. In the accompanying diagram, AB is the diameter of circle O and chord CD intersects diameter AB at E.

2. In the diagram below, ΔABC is inscribed in circle P. The distances from the center of circle P to each side of the triangle are shown.

Answer the following questions:
What is the length of the radius of the circle shown?
What is the length of AB and AC?

3. In the following circle, QT is a diameter that goes through point S.

Find the measure of OP, QR and QT. Justify your answers.

It looks like you've provided a set of geometry questions involving circles, chords, diameters, and triangles inscribed in circles. Let's break down and analyze each problem:

### 1. Diagram with Diameter AB and Chord CD Intersecting at E:
- **Given**: AB is the diameter of circle \( O \) and chord \( CD \) intersects the diameter at point \( E \).
- **To Solve**:
  - Since \( AB \) is the diameter, by the **Thales' theorem**, \( \angle CEB = 90^\circ \) if \( C \) and \( D \) lie on the circumference.
  - You can use properties of the circle and similar triangles to find specific lengths, angles, or other required elements.
  
### 2. Triangle ABC Inscribed in Circle P:
- **Given**: \( \Delta ABC \) is inscribed in circle \( P \) with distances from the center \( P \) to each side of the triangle provided.
- **To Solve**:
  - The length of the radius \( r \) of the circle can be determined if any of the distances or the circumradius formula is given.
  - The length of \( AB \) and \( AC \) might require knowing the triangle’s circumradius \( R \) and applying the Law of Sines or another relevant formula.
  
### 3. Circle with Diameter QT Passing Through Point S:
- **Given**: \( QT \) is a diameter passing through point \( S \).
- **To Solve**:
  - The measure of \( OP \) can be found if \( O \) is the center of the circle and \( P \) is some point on the circle or inside.
  - The length of \( QR \) and \( QT \) depends on the circle's properties and whether \( QR \) is a chord, radius, or another element.

### To move forward:
1. **What is the radius of the circle in Problem 2?**
2. **What are the distances provided from the center \( P \) to each side of the triangle in Problem 2?**
3. **Are there any specific lengths or angles given in Problems 1 and 3?**
4. **Can you upload a diagram for more precise analysis?**
5. **What is the position of point \( S \) relative to the center in Problem 3?**

**Tip:** Remember, when dealing with inscribed angles and chords in a circle, key theorems like the Inscribed Angle Theorem and properties of cyclic quadrilaterals are often useful.

Feel free to provide more details or upload a diagram for further assistance!

Explore detailed solutions to geometry problems involving circles and triangles, including determining circle radius, lengths of diameters and chords, and applications of Thales' Theorem and Law of Sines. Perfect for high school students studying circle geometry.

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