VectorPaint three-four-five square dissection: check outer49 and central25

English translation of the Chinese original, translated on 2026-11-10. Original publication dates, attribution, code and version notes are retained.

Four right triangles around a gap do not automatically make that gap square. Specify joins and check edges, adjacent directions and areas separately. Use manual paths and external arithmetic for this original dissection, without automatic tessellation, solving or theorem proving, and distinguish ideal dimensions from drawing precision.

Define the seven-by-seven outer square

Use corners(0, 0), (7, 0), (7, 7), (0, 7), side 7, area 49. Scale uniformly by 80 pixels/unit and translate into margins, with canvas-downward y throughout. Draw a pale outer reference before triangles rather than guessing its side afterward. Keep title/units outside and exclude stroke width from ideal length. This is not a physical template or an assumed pixel-to-centimetre conversion; printing/cutting requires separate checks.

Place four boundary joins in order

Set boundary joins A(3, 0), B(7, 3), C(4, 7), D(0, 4). Each side splits into 3 and 4 along its own orientation, not always three from page-left. Preserve outer corners separately. A–D define the central region, not the outer centre. Repair off-edge coordinates rather than enlarging the square. Placement matters to non-overlapping corner triangles; equal areas alone do not establish proper positioning.

Use three and four as each corner’s perpendicular legs

Join each outer corner to neighbouring joins, forming four right triangles with legs 3, 4 and area 3×4/2=6, total 24. Avoid accidental 3/3 or 4/4 legs. Temporary different fills aid dimension checking without changing areas. Hide central fill and verify outer corners and inner edgesAB, BC, CD, DA. Repair shared points for cracks instead of thickening strokes. Duplicated/rotated pieces still need endpoint checks rather than visual similarity alone.

Check central equal sides and right angles

Edge displacements are(4, 3), (−3, 4), (−4, −3), (3, −4). Each squared length is 25, so length 5. Adjacent dot product 4×(−3)+3×4=0 establishes perpendicularity, with the pattern repeating. Thus the centre is a side 5 square. Area 25 alone would not prove shape. State equal-side and right-angle evidence; even without dot-product terminology retain explicit right-triangle orientation checks rather than “looks square”.

Compare subtraction with direct area calculation

Outer 49 minus four triangles 24 leaves 25; the independently checked side 5 centre also has area 25. State the no-overlap/no-other-gap condition required by subtraction. This verifies the numerical 3–4–5 instance, not a software proof for every right triangle. Area uses square units; 3, 4, 5, 7 are lengths, while 49/24 must not be labelled edge lengths.

Test which conditions fail when moving one join

In a wrong copy move A from(3, 0) to(4, 0), leaving others fixed. Two central edges/corner triangles change, so four identical 3/4 triangles can no longer be claimed. Mark and recalculate affected edges rather than fixing the centre label at 25. Preserve the correct view; the counterexample tests shape and area together. Recheck other joins after restoration to detect accidental movement. No exhaustive classification of dissections is required; this layout depends on its shared coordinates.

Inspect joins and boundaries before styling

Save and open a PNG, checking four corners, central edges and 7×7 extent. Hide unwanted manual grids actively. For visible gaps inspect endpoints/closure instead of recolouring everything to conceal them. Fine structural strokes do not enter ideal area. A monochrome version should retain positional traceability without requiring four colours. Open other formats separately rather than validating all output from a canvas screenshot.

Troubleshoot outer square, joins, pieces and centre

For correct residual area but a skewed centre revisit side/angle checks; for a non 24 triangle total revisit legs; for a non 7 outer side inspect scaling. Recheck neighbouring pieces after repairs rather than editing one label. Preserve coordinates, four sixes, outer 49, central 5/25 and counterexample. This explains a dissection, not automatic assembly, formal general proof or physical paper precision.