VectorPaint quadrilateral midpoints: verify a parallelogram and half-area seventeen

Joining side midpoints of a convex quadrilateral gives a parallelogram, not necessarily a rectangle. Fix an original area-34 quadrilateral in VectorPaint, construct four midpoints and verify area 17. Check midpoint, parallelism and half-area independently without automatic geometry recognition.

Fix a convex quadrilateral and vertex order

Set A=(0,0),B=(8,0),C=(6,4),D=(0,6), joined in that order. On an approximately 1700×1400 canvas use 100 px/unit and screen origin (300,1000), giving B=(1100,1000), C=(900,600), D=(300,400). Separate original, diagonals, midpoint figure and notes. Keep it simple/convex; changing C to (6,-4) changes the example. Record mathematical versus screen coordinates, subtracting positive y; units are not millimetres.

Average each edge’s endpoints separately

Compute E=(4,0),F=(7,2),G=(3,5),H=(0,3) from adjacent endpoint averages, not one average of all four vertices. Place them on their edges before joining in order. For a misplaced F check B/C versus B/D. For G=(3,2), restore (4+6)/2=5. Move crowded labels without moving midpoint dots off their edges.

Explain both parallel pairs through diagonals

AC has vector (6,4); EF and HG are (3,2), half of AC. BD is (-8,6); EH and FG are (-4,3), half of BD. Thus opposite sides are parallel and equal. Preserve vector direction: GH=(-3,-2) is still parallel to EF despite opposite signs. Pale diagonals are manual aids, not automatic vector reports or locked midpoint constraints.

Calculate the original area thirty-four first

Split along AC. Triangle ABC has base 8 and height 4, area 16; ACD has vertical base AD=6 and perpendicular distance 6, area 18. Total 34. Heights are perpendicular distances, not AC length. The two triangles cover this convex figure without overlap. Units are squared invented units; scale 100 px/unit would multiply area by 10000, not 100. Keep this ledger in mathematical units.

Calculate midpoint area seventeen, not a visual half

From E, adjacent vectors EF=(3,2), EH=(-4,3) give absolute determinant 3×3−2×(-4)=17, half of 34. This follows arithmetic, not colour coverage. Alternatively, four corner triangles have areas 6,4,2.5,4.5, totalling 17; subtracting from 34 leaves 17. Keep the same coordinates/order for both methods rather than borrowing an answer after changing the figure.

Do not upgrade a parallelogram into a rectangle

EF·EH=3×(-4)+2×3=-6, not zero, so adjacent sides are not perpendicular and the inner figure is no rectangle. Axis-aligned outer edges justify no inner right-angle marks. Use parallel markers first. Retain corner regions in the native work rather than masking inconvenient boundaries. Half-area does not halve each corresponding outer side; the half-length relations concern inner sides and original diagonals.

Troubleshoot midpoint and area-direction errors

For failed parallelism recalculate midpoints before dragging curves into place. For area 35 check heights/arithmetic; for -17 take the absolute oriented determinant. Opposite vector signs do not defeat parallelism. Correct one issue at a time, updating the entire diagram after origin/scale changes. Inspect export of ordinary diagonal guides separately. This is static manual arithmetic, not a built-in area meter or verified proof system.

Reopen and verify four midpoints and both area ledgers

Save native work, export PNG and reopen. Check ordered E-F-G-H on original sides, then coordinates and areas 34/17 enlarged. Move labels, not points; adjust opacity/stroke if inner fill hides outer edges. Reopen separately editable figures and retain diagonals plus both area checks. Explain parallelism through half-diagonal vectors, area through calculation and why no rectangle follows. Appearance, dragging and one export are not universal automatic proofs.

Related tutorials and sources

VectorPaint parallelogram dissections: fix base and perpendicular height before area; VectorPaint triangle medians: check midpoints and the two-thirds centroid position.

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.

Geometry: Euclid VI.2 links proportional triangle sides and parallel lines. Original coordinate differences and area arithmetic independently verify this midpoint construction; screenshot appearance is no proof.