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The image shows a geometry problem that involves a circle and two triangles inscribed within it. Here's the transcription of the problem:

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**Question:**

1. The points \( M \) and \( N \) are the midpoints of \( AB \) and \( AC \) respectively.  
    All perpendiculars are perpendiculars from \( AB \) to the opposite side.

   (a) Prove that \( \triangle OAB \) is an isosceles right triangle.  
   (b) Prove that \( OM = ON \).

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### Explanation for Part (a):

To prove that \( \triangle OAB \) is an isosceles right triangle, we need to show that:

1. \( OA = OB \): This implies that \( O \) is the center of the circle (since radii of the circle are equal).
2. One angle, likely \( \angle OAB \) or \( \angle OBA \), must be \( 90^\circ \) for it to be a right triangle.

Since \( O \) is the center of the circle, and both \( OA \) and \( OB \) are radii of the circle, we know they are equal, making \( \triangle OAB \) isosceles. The right angle comes from the perpendicular condition in the problem statement.

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### Explanation for Part (b):

To prove that \( OM = ON \), we use the fact that \( M \) and \( N \) are the midpoints of \( AB \) and \( AC \), respectively. Since \( O \) is the center and the triangle is symmetric, the distances from \( O \) to the midpoints should be equal, based on symmetry.

Would you like more details on these proofs?

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### Related Questions:

1. How do you determine the center of a circle given multiple geometric constraints?
2. What are the properties of an isosceles triangle in a circle?
3. How can you use the midpoint theorem to prove symmetry in a triangle?
4. Why are the radii of a circle always equal?
5. What other shapes can be formed from similar triangle properties in circles?

**Tip**: In geometric problems involving circles, always look for symmetry and equal radii to simplify your approach.

The points M and N are the midpoints of AB and AC respectively. (a) Prove that triangle OAB is an isosceles right triangle. (b) Prove that OM = ON.

Explore a geometry problem involving an isosceles right triangle inscribed in a circle, and learn how to prove symmetry using midpoints. This detailed solution covers key concepts such as isosceles triangles, the Pythagorean theorem, and symmetry in geometry. Ideal for students in Grades 9-12.

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