Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.3 AIME, harder Prove it United States

Problem:

A given line passes through the center $O$ of a circle. The line intersects the circle at points $A$ and $B$ . Point $P$ lies in the exterior of the circle and does not lie on the line $A B$ . Using only an unmarked straightedge, construct a line through $P$ , perpendicular to the line $A B$ . Give complete instructions for the construction and prove that it works.

Solution

Solution:

1. Draw a line from $P$ to $A$ , intersecting the circle at $C$ .
2. Draw a line from $P$ to $B$ , intersecting the circle at $D$ .
3. Draw lines $A D$ and $B C$ , and let $E$ be their point of intersection.
4. Draw a line from $P$ through $E$ ; this will be the desired perpendicular line.

This works because $A D \perp P B$ and $B C \perp P A$ ; hence $A D$ and $B C$ are altitudes of triangle $A P B$ . It is well known that the three altitudes of a triangle intersect in a point, so $E$ is the intersection of all three altitudes. It follows that the line through $P E$ is an altitude.

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