VectorPaint cyclic quadrilateral: preserve circular order before checking opposite angles

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.