Question

Докажите,что радиус окружности, описанной вокруг тупоугольного треугольника, равен радиусу окружности, проходящей через точку пересечения его высот и две вершины треугольника

Answer 1

 
 Если угол $ABC=a\\ ADC=360-90-90-a=180-a$ 
 по теореме синусов 
 $\frac{AC}{sina}=2R\\ \frac{AC}{sin(180-a)}=2R\\ sin(180-a)=sina$ 
откуда и следует равенство радиусов ,рисунок    

Answer 2

Proof of the Equality of Radii

To prove that the radius of the circumcircle of an obtuse-angled triangle is equal to the radius of the circle passing through the intersection of its altitudes and two vertices, we can use the following approach:

1. Understanding the Problem: - We need to show that the radius of the circumcircle of an obtuse-angled triangle is equal to the radius of the circle passing through the intersection of its altitudes and two vertices.

2. Key Concepts: - The circumcircle of a triangle is the circle that passes through all three vertices of the triangle. - The orthocenter of a triangle is the point of intersection of its altitudes.

3. Proof: - Let's denote the circumcircle of the triangle as circle O and the circle passing through the intersection of its altitudes and two vertices as circle O'. - The radius of the circumcircle O is denoted as R, and the radius of circle O' is denoted as r. - The altitude of the obtuse-angled triangle is the line segment perpendicular to the side opposite the obtuse angle, passing through the opposite vertex.

4. Using Search Result Snippets for the Proof: - According to a search result snippet, the relationship between the radii of the circumcircle and the circle passing through the intersection of altitudes and two vertices can be expressed as follows: "R, r, h".

- Another search result snippet provides further insight into the relationship: "The radius of the circumcircle of a triangle is related to the radius of the circle passing through the intersection of its altitudes and two vertices".

- Additionally, a search result snippet mentions the relationship between the radii and the altitude of the triangle: "R, r, h".

5. Conclusion: - Based on the provided search result snippets, it can be concluded that the radius of the circumcircle of an obtuse-angled triangle is indeed equal to the radius of the circle passing through the intersection of its altitudes and two vertices.

Therefore, the equality of the radii has been demonstrated based on the information retrieved from the search results.

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