Angle Bisector Equidistance Theorem

An animated proof demonstrating that any point on an angle bisector is equidistant from the two sides of the angle, and conversely, any point equidistant from the two sides lies on the angle bisector.

Keywords: angle bisector, equidistant, congruent triangles, AAS congruence, HL theorem

Prerequisites: triangle-congruence · Difficulty: beginner

An angle bisector splits an angle into two equal halves. We show it is exactly the set of points equidistant from the angle's two sides:

Forward: a point on the bisector is equidistant from both sides
Converse: a point equidistant from both sides lies on the bisector

Strategy: drop perpendiculars from the point $P$ to each side, meeting at $D$ and $E$ , then compare triangles $△ P A D$ and $△ P A E$ — AAS for the forward direction, Hypotenuse-Leg for the converse.

Step 1: Part 1: Setup

We start with angle $∠ B A C$ and its angle bisector.

Let $P$ be any point on the bisector. We'll prove that $P$ is equidistant from both sides — that is, the perpendicular distances $P D$ and $P E$ are equal.

Step 2: Proving $P D = P E$ by AAS

Form triangles $△ P A D$ and $△ P A E$ . We prove they're congruent:

$∠ P A D = ∠ P A E$ (P is on the bisector)
$∠ P D A = ∠ P E A = 90°$ (perpendiculars)
$P A = P A$ (common side)

By AAS congruence, $△ P A D ≅ △ P A E$ , so $P D = P E$ .

Step 3: Part 2: The Converse

Now we prove the reverse: if a point $P$ is equidistant from both sides ( $P D = P E$ ), then $P$ must lie on the angle bisector.

Step 4: Proving $P$ is on the Bisector by HL

Form right triangles $△ P A D$ and $△ P A E$ . We prove congruence:

$P D = P E$ (given)
$P A = P A$ (common hypotenuse)
Both have right angles

By the Hypotenuse-Leg theorem, $△ P A D ≅ △ P A E$ .

Therefore $∠ P A D = ∠ P A E$ , so ray $A P$ bisects the angle — $P$ is on the bisector!

Step 5: Conclusion

We've proven both directions:

Forward: Points on the bisector are equidistant from both sides
Converse: Points equidistant from both sides lie on the bisector

The angle bisector is the locus of all points equidistant from the two sides!

We've proven both directions:

Forward: Points on the bisector are equidistant from both sides (by AAS congruence)
Converse: Points equidistant from both sides lie on the bisector (by HL theorem)

The angle bisector is exactly the locus of points equidistant from the two sides. This is why angle bisectors are so important for finding the incenter! ∎

Notes

Why This Matters

This property is the foundation for the incenter of a triangle. Since the incenter lies on all three angle bisectors, it must be equidistant from all three sides — which is why it's the center of the inscribed circle!