8,9

In a triangle with sides 6, 10, and 12, an inscribed circle is drawn. A tangent to the circle intersects the two larger sides.

Find the perimeter of the cut-off triangle.

#
The distance from the vertex of a triangle to the nearest point of tangency with the inscribed circle is equal to the difference between the semiperimeter and the opposite side.

## Solution

Let $K$ be the point of tangency of the inscribed circle in triangle $ABC$ with side $AB$ ($AB=10, AC=12, BC=6$). If $p$ is the semiperimeter of the triangle, then $AK = p - BC = 14 - 6 = 8$, and the length of segment $AK$ is equal to the semiperimeter of the cut-off triangle.

![](https://cdn.mathpix.com/cropped/2024_05_06_0ef0b80dacb6d98eff5ag-08.jpg?height=434&width=283&top_left_y=835&top_left_x=888)

## Answer

## Problem