A bisector depends on rays and the selected region
An angle bisector is a ray from the vertex that divides a selected angle into two equal angles. For direction vectors u and v, this angle bisector tool normalizes both before combining them, so endpoint distance does not distort direction. Minor and reflex are explicit choices: selecting reflex reverses the internal minor direction and halves 360° minus the minor angle.
Numeric mode sets a horizontal first ray and rotates the second. Coordinate mode treats P as the vertex, Q as the first endpoint, and R as the second. Dragging any point updates those same coordinates. Grid and snap affect interaction only. Construction mode adds equal-radius compass arcs as a visual recreation of the classical steps.
Three ways to define and divide an angle
- Use Numeric to enter a selected angle, or Coordinates to edit P, Q, and R. You can also drag each labeled point directly on the canvas.
- Choose minor or reflex before reading the highlighted region. Reflex is never inferred from a smaller-looking wedge.
- Choose internal for the selected region or external for the perpendicular difference-vector direction. Opposite rays require one explicit perpendicular choice.
- Enable construction steps to show the vertex arc, equal-radius arcs from the intersection points, and the line through their intersection.
- Copy values, share URL state, or export SVG/PNG. Exported graphics include the current geometry rather than account or score data.
Normalized vector method
- Create nonzero vectors u = Q − P and v = R − P, then normalize them to û = u/|u| and v̂ = v/|v|.
- The minor internal direction is normalize(û + v̂). For the selected reflex region, the opposite ray is used so the two displayed subangles each equal half the reflex turn.
- The external direction is normalize(û − v̂) and is perpendicular to the ordinary internal bisector when neither construction is degenerate.
- A zero-length ray has no direction and is rejected. When rays are opposite, û + v̂ is zero and two perpendicular 180° bisectors are equally valid, so a choice is required.
- The angle is obtained from acos(clamp(û · v̂)). The clamp handles floating-point spillover, while SVG coordinates remain separate from mathematical units.
Worked vector and construction examples
- For rays at 0° and 60°, normalized vectors sum toward 30°. The internal line creates two 30° angles.
- For perpendicular rays (1,0) and (0,1), the internal direction normalize(1,1) points at 45°. The external normalize(1,−1) is 90° from it.
- With P = (2,2), Q = (5,2), and R = (2,6), unequal ray lengths still produce a 90° angle and a 45° bisector because each vector is normalized first.
- Selecting the reflex region for a 90° minor angle creates a 270° selected turn, so each reflex subangle is 135°.
Compass-and-straightedge sequence
Draw an arc centered on the vertex so it crosses both rays. From those two intersection points, draw equal-radius arcs that meet inside the selected region. A straight line from the vertex through that meeting point is the internal bisector. Equal radii create congruent triangles, which establish the equal subangles.
- One vertex-centered arc.
- Two equal-radius intersection arcs.
- A line through the vertex and arc intersection.
Practice without accounts
Practice creates a random target angle and lets you place a candidate bisector by direction. Difficulty changes the tolerance. Feedback reports angular deviation, not a public score, and the next problem can be generated locally.
Screen construction and exact geometry
The vector answer is numerically calculated; the compass overlay is an instructional screen construction, not evidence of physical compass accuracy. Pointer snapping and pixel export introduce display resolution. Coordinates are unitless, and the tool does not become a triangle angle-bisector-theorem calculator or establish segment-length ratios on an opposite side.
Points and options remain in the browser and may be encoded into the share URL. No account or leaderboard is used.