Burnt Rainbow

55

Fig. 4.

From the similarity of triangles AON and AMK we find an exception:

$$\frac{AO}{MO} = \frac{AN}{KN} = \frac{AN}{CN} \cdot \frac{1+m}{m} = \frac{n}{m} (1+m),$$

and similarly

$$\frac{BO}{NO} = \frac{m}{n} (1+n).$$

Exercises

1. The edge of regular triangle ABCD is $a$. Find the radius of the sphere passing through the vertices $A, B$, the center of the face ACD and the middle of the edge BC.

2. On the base AC of equilateral triangle ABC, point D is located so that AD=a, CD=b. The circles written in triangles ABD and BCD touch line BD at points M and N, respectively. Find the length MN.

3. Triangle ABC is inscribed in a circle. The distances from points A and C to the line tangent to the circle at point B are $a$ and $c$ respectively. Find the altitude of triangle ABC drawn from vertex B.

4. In an acute triangle, the two heights are equal to 3 cm and $2\sqrt{2}$ cm, respectively, and the point of their intersection divides the triangular height in a ratio of 5:1, counting from the vertex of the triangle. Find the area of ​​the triangle.

5. At the base of a right triangular sign lies a right triangle, the legs of which are equal to $a$ and $b$. Some plane intersects all the side edges of the plane so that in cross-section a regular triangle is obtained. Determine the side of this triangle.

6. In a regular quadrangular pyramid SABCD, the height is equal to the diagonal of the base ABCD. A plane is drawn through vertex A parallel to line BD and tangent to the ball inscribed in the pyramid. Find the ratio of the cross-sectional area to the area of ​​the base of the pyramid.