# VectorPaint cyclic quadrilateral: preserve circular order before checking opposite angles
Language: en
Canonical: [VectorPaint cyclic quadrilateral: preserve circular order before checking opposite angles]()
中文：[中文 HTML]()
English: [English HTML]()
Published: 2026-11-15T00:00:00+08:00
Modified: 2026-11-15T10:05:49+08:00
Author: Bison
Use four circular positions to check 105+75 and 90+90, keeping cyclic and crossed polygons distinct.
English translation of the [Chinese original](), translated on 2026-11-15. Original publication dates, attribution, code and version notes are retained.
Four points around a circle do not make adjacent angles supplementary. Build a convex cyclic quadrilateral in VectorPaint, check opposite pairs summing to 180°, then diagnose crossed connections and wrong pairings. Use manually arranged circles, points and paths without automatic cyclic recognition or angle measurement.
## Fix one circle and mathematical angle direction
Use an approximately 1700×1600 canvas with O as mathematical origin and radius 4, optionally 100 px/unit for diameter 800. Measure counterclockwise from positive x, converting downward screen y. Separate circle, points, sides and annotations. Restore equal width/height before placing points on a distorted ellipse. Radius units are invented, not millimetres. Moving text must not move its point.
## Place points at zero, sixty, one-eighty and two-seventy degrees
Place A,B,C,D at 0°,60°,180°,270° in circular order, with mathematical coordinates (4,0),(2,2√3),(-4,0),(0,-4); B’s y≈3.464. At screen centre (800,800), positions are approximately (1200,800),(1000,453.6),(400,800),(800,1200). Keep x/y pairs distinct and do not put a negative mathematical y above the screen centre. Label approximate placement if necessary, retaining mathematical angles without claiming automatic constraints.
## Connect ABCD in circular order
Join A-B-C-D-A, rather than the crossed A-C-B-D order. A pale AC diameter is a guide, not a fifth polygon edge or a new vertex at an intersection. Hide the circle to inspect a simple closed outline, then restore it and check all four points. A centre-crossing line may be a guide; the error is changing polygon order while retaining convex-angle claims. Keep external labels away from actual endpoints and angle arcs.
## Calculate intercepted arcs excluding the vertex
Adjacent arcs are 60°,120°,90°,90°, totalling 360°. Angle A intercepts BCD=210°, giving 105°; angle C intercepts DAB=150°, giving 75°. Identify the arc before halving it, never substitute small arc AB or copy a central angle. Use declared point order, not label placement. Angle A uses AB/AD, and angle C uses CB/CD.
## Check the other opposite pair through the diameter
B intercepts ADC=180° and D intercepts ABC=180°, so each is 90°. Check A+C=180° and B+D=180°. Adjacent A+B=195° need not be supplementary. Four right angles would wrongly extend a diameter property to every vertex; this is no rectangle. Total 360° is an additional check but cannot detect swapped 105°/75° labels.
## Separate angle marks from reference arcs
Put small interior angle arcs and values at vertices, grouping A/C and B/D by both colour and names. Keep circular arc annotations on guides. Hide selected guides for a cleaner final while retaining calculations. Do not label an exterior arc 105° as an interior angle. Move crowded text without moving points. Stroke width, arrows and screenshot appearance do not define angle values; retain data after scaling.
## Diagnose crossed order and wrong pairings separately
In a separate copy, cross connections, identify changed order/self-intersection and undo; do not apply this convex-angle convention there. A second control keeps geometry but falsely pairs A/B, producing 195°. Change these errors separately. Rebuild an off-circle point from angle/radius rather than editing 180° to fit. Checks concern the declared construction, not automatic cyclic classification of arbitrary quadrilaterals.
## Reopen and verify order and both 180-degree pairs
Save native work, export PNG and reopen. Read circular ABCD order and sides, then interior values A105,B90,C75,D90 enlarged. Label exported AC as a diameter guide or retain it hidden in the original. Reopen separately editable labels/points and record numeric versus approximate placement. Deliver arc total, both opposite sums and the adjacent-angle counterexample, without claiming precision testing or extending conclusions to noncyclic, crossed or degenerate quadrilaterals.
## Related tutorials and sources
[VectorPaint two tangents from an external point: check right angles and equal lengths](); [VectorPaint inscribed regular hexagons: verify six sides and six equilateral triangles]().
Sources checked on 15 November 2026: the [official VectorPaint App Store listing]() and [Apple public lookup]() show version 1.0.8, requiring iOS / iPadOS 17.0 or later. Draw manually with published shapes, paths, nodes, layers and text. Conditions, scales and layouts are original exercise arrangements; check installed controls. Keep editable native work and PNG previews. SVG/PDF may retain any bitmap content used. External calculations and static diagrams are not new automatic proof, statistics or solver features. No completed device, physical-object or universal export testing is claimed.
Theorem: [Euclid III.22]() gives supplementary opposite angles in a cyclic quadrilateral. Point choices and calculations are original, not an application to every noncyclic quadrilateral.