8,9

[ Two tangents drawn from one point ]

In an isosceles triangle with a lateral side of 100 and a base of 60, a circle is inscribed. Find the distance between the points of tangency located on the lateral sides.

#
The segment connecting the points of tangency of a circle with the lateral sides of an isosceles triangle cuts off a triangle similar to the given one.

## Solution

Let \( M \) and \( N \) be the points of tangency of the circle with the lateral sides \( AB \) and \( AC \) of the isosceles triangle \( ABC \), and \( K \) be the point of tangency with the base. Then, \( AM = AB - MB = AB - BK = 100 - 30 = 70 \).

Since \( AM = AN \), triangles \( AMN \) and \( ABC \) are similar. Therefore, \( MN : BC = AM : AB \). Consequently, \( MN = BC \cdot AM / AB = 42 \).

![](https://cdn.mathpix.com/cropped/2024_05_06_d3cd76ad0dc53ae543bdg-29.jpg?height=375&width=460&top_left_y=533&top_left_x=799)

## Answer

42.