Library / Harvard-MIT Mathematics Tournament /107 of 352

Harvard-MIT Mathematics Tournament

Geometry Difficulty 4.9 AIME Prove it United States

Problem:

Let $ABC$ be an isosceles triangle with apex $A$ . Let $I$ be the incenter. If $AI = 3$ and the distance from $I$ to $BC$ is $2$ , then what is the length of $BC$ ?

Solution

Solution:

Let $X$ and $Y$ be the points where the incircle touches $AB$ and $BC$ , respectively. Then $AXI$ and $AYB$ are similar right triangles. Since $I$ is the incenter, we have $IX = IY = 2$ . Using the Pythagorean theorem on triangle $AXI$ , we find $AX = \sqrt{5}$ . By similarity, $AY / AX = BY / IX$ . Plugging in the numbers given, $5 / \sqrt{5} = BY / 2$ , so $BY = 2\sqrt{5}$ . $Y$ is the midpoint of $BC$ , so $BC = 4\sqrt{5}$ .

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.