Perpendicular Bisector Basics
A perpendicular bisector is a line that crosses a segment at its midpoint and meets the
segment at a right angle. If a segment has endpoints \(A(x_1,y_1)\) and \(B(x_2,y_2)\), its midpoint is
\[
M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right).
\]
“Bisector” means it splits the segment into two equal lengths, so \(|AM|=|MB|\). “Perpendicular” means the bisector’s
direction forms a \(90^\circ\) angle with the segment’s direction.
The most important geometric meaning is that the perpendicular bisector is an equidistant locus.
A locus is the set of all points satisfying a condition. Here the condition is:
\[
PA = PB,
\]
meaning every point \(P\) on the perpendicular bisector is the same distance from \(A\) and \(B\).
This is why perpendicular bisectors appear so often in construction problems: whenever you need a point that is equally
far from two locations, you look on the perpendicular bisector of the segment joining them.
Direction and Special Cases
A segment’s direction vector is \(\overrightarrow{AB}=(\Delta x,\Delta y)=(x_2-x_1,\;y_2-y_1)\).
A perpendicular direction can be formed by rotating the vector \(90^\circ\), for example
\[
\mathbf{p}=(-\Delta y,\;\Delta x).
\]
A line through \(M\) in the direction \(\mathbf{p}\) is perpendicular to \(\overline{AB}\). In coordinate geometry, slopes
summarize this idea:
- If \(\Delta x=0\), the segment is vertical, so the perpendicular bisector is horizontal.
- If \(\Delta y=0\), the segment is horizontal, so the perpendicular bisector is vertical.
- Otherwise, if \(m_{AB}=\Delta y/\Delta x\), then a perpendicular slope is \(m_\perp=-\Delta x/\Delta y\).
This tool shows these properties but intentionally avoids printing a full equation form, since the equation-focused locus
calculators come later.
Why It Matters: Circumcenter Preview
In any triangle, the perpendicular bisectors of the three sides meet at a single point called the circumcenter.
That point is equidistant from all three vertices, so it is the center of the triangle’s circumcircle. This is a powerful
idea: “equal distances” becomes a simple intersection of perpendicular bisectors.
Quick Glossary
Segment: the finite piece between two endpoints. Midpoint: halfway point of a segment.
Perpendicular: meeting at \(90^\circ\). Bisector: splits into equal parts.
Locus: set of points satisfying a condition. Circumcenter: intersection of perpendicular bisectors in a triangle.
