Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.7 AIME Prove it United States

Problem:

A line $l$ and two points $A$ and $B$ are given in a plane in such a way that $A$ belongs to $l$ but $B$ doesn't. Construct the circle $k$ that passes through $B$ and touches $l$ at the point $A$ .

Solution

Solution:

Let $m$ be the bisector of the segment $A B$ . The center of the circle $k$ has to belong to $m$ . Similarly, since the circle has to be tangent to $l$ at the point $A$ its center has to be located on the line $a$ perpendicular to $l$ that passes through $A$ .

The lines $m$ and $a$ are easy to construct and their intersection is the center $O$ of the circle. Now we have the center and the point $B$ hence the circle is determined.

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