# VectorPaint perpendicular bisectors: check equal-radius intersections and midpoint

Language: en
Canonical: [VectorPaint perpendicular bisectors: check equal-radius intersections and midpoint](<https://allluckly.cn/en/guides/vectorpaint-perpendicular-bisector-equal-arcs/>)
中文：[中文 HTML](<https://allluckly.cn/guides/vectorpaint-perpendicular-bisector-equal-arcs/>)
English: [English HTML](<https://allluckly.cn/en/guides/vectorpaint-perpendicular-bisector-equal-arcs/>)
Published: 2026-11-09T00:00:00+08:00
Modified: 2026-11-09T09:37:10+08:00
Author: Bison

Construct intersecting equal-radius circles for a horizontal segment and check the line through both intersections, excluding a degenerate single-point contact.

English translation of the [Chinese original](<https://allluckly.cn/guides/vectorpaint-perpendicular-bisector-equal-arcs/>), translated on 2026-11-09. Original publication dates, attribution, code and version notes are retained.

Numerical midpoints are useful, but equal-radius arcs provide another check. Draw two circles and their intersections, join the perpendicular bisector and test an insufficient-radius counterexample. This manual geometric explanation assumes no dedicated construction-proof tool, constraint solver or automatic snapping to theoretical intersections.

## Define endpoints and coordinate convention

Use a1200×900 canvas with A(300,450), B(700,450), a horizontal segment of length400. Join and label endpoints under rightward-x/downward-y coordinates. Record candidate midpoint(500,450) externally without using it to draw the final perpendicular immediately. Keep endpoint marks distinct from text centres. Lengths are exercise coordinates, not physical construction precision; viewing zoom does not change theoretical endpoint distance.

## Use radius250 for both circles

Construct radius250 circles centred on A and B, both diameter500. Check with available visible size controls or separate bounds: A’s top-left(50,200), B’s(450,200), each500×500. Do not input radius for one and diameter for the other. Preserve whole-circle checks before optionally showing only arcs near intersections. Isolated intersecting curves without centre/radius records do not establish equality. Nonuniformly scaled ellipses also fail this requirement.

## Identify one intersection above and one below

Intersections are(500,300) and(500,600). Both have horizontal distance200 from each endpoint and vertical offset150 from the segment, giving distance250 by the150–200–250 right triangle. Check circles against these coordinates before labelling C/D. Text near an intersection is not the intersection itself, and overlapping stroke pixels do not establish one precise point. If controls permit only approximation, distinguish illustrative placement from the external calculation.

## Join intersections and then check the midpoint

Join C/D to form vertical x=500, crossing AB at(500,450). The crossing is200 from each endpoint, so the line both bisects and is perpendicular. State both properties instead of judging a cross shape alone. Pale circles can remain behind a distinct final line. Moving A or B requires reconstructing centres, intersections and bisector, not just the midpoint label. Ordinary objects and grouping do not automatically maintain these geometric conditions.

## Test too-small and exactly-half radii separately

Save separate controls: radius150 gives a radius sum300 less than endpoint separation400, so no intersection; radius200 gives only midpoint tangency, not two distinct upper/lower intersections. Touching the correct midpoint cannot determine the line through two intersections used here. Preserve the250 case and identify all views rather than overlaying unexplained alternatives. Radius must exceed half the segment length, not merely match between circles. Do not stretch small circles into ellipses to manufacture intersections.

## Retain traceability when displaying only arcs

For a cleaner final, duplicate the check and show arcs near intersections while retaining centres and equal-radius labels. Keep C/D and their actual crossings. These arcs are not unrelated decorative curves; node edits require checking against full circles again. If splitting circles safely is unavailable, keep pale complete circles rather than seeking hidden commands. Simplification changes presentation, not conditions. Name full-circle and simplified versions distinctly so short arcs retain provenance.

## Check coordinates and appearance after export

Save and open PNG/PDF output, checking nonuniform circle scaling, displaced intersections and labels obscuring centres. Normal size should explain sequence; enlargement checks stroke endpoints without implying infinite precision. Hide unwanted manual coordinate grids; verify native-grid export separately. For a missing intersection inspect cropping before changing radius. For unequal circles inspect dimensions instead of compensating with stroke thickness.

## Separate the construction from its evidential scope

Retain separation400, radius250, both intersections, midpoint and degenerate controls. For a tilted bisector verify that intersections belong to the same circles; for a perpendicular that misses midpoint check centres/radii. Attractive graphics do not constitute software formal proof or intelligent construction objects. This is one reproducible instance with explicit conditions, requiring reconstruction after coordinate changes, not verification of all geometry precision or physical drawing instruments.

## Related tutorials and sources

[VectorPaint grids and guides: composition references for icons and posters](<https://allluckly.cn/en/guides/vectorpaint-grid-guides/>), [VectorPaint triangle medians: check midpoints and the two-thirds centroid position](<https://allluckly.cn/en/guides/vectorpaint-triangle-median-centroid/>).

Checked on November 9, 2026: VectorPaint public version 1.0.8, requiring iOS / iPadOS 17.0 or later. Capability scope follows the [official App Store listing](<https://apps.apple.com/cn/app/id1587272880>) and [Apple public version lookup](<https://itunes.apple.com/lookup?id=1587272880&country=cn>). This is an original exercise to perform independently; use controls in your installed version. Manual construction and external records do not imply new automatic analysis features. No step-by-step physical-device, private-file or all-output-environment testing is claimed.