VectorPaint angle-bisector diagram: six to nine gives four to six, not equal base halves

Bisecting a vertex angle need not halve the opposite side. Draw a six-nine-ten triangle in VectorPaint, locating its internal bisector at base lengths four and six and comparing midpoint five. Calculate ratios before arranging points, lines and labels; no automatic bisector or grading tool is assumed.

Bind side names to lengths first

On an approximately 1900×1600 canvas, set BC=10 with B left, C right and 60 px per unit, giving 600 px. Screen positions may be B=(600,1000), C=(1200,1000). Bind AB=6 and AC=9 to letters. Separate outline, guides and labels, leaving room below. Keep mathematical lengths distinct from pixels, not millimetres or claimed device measurements. Drawing error does not justify silently changing the declared sides.

Place the vertex with two reference circles

Draw reference circles centred at B and C with radii 6 and 9, hence diameters 720 and 1080 px. Select the upper intersection A and join sides; label approximate placement if no exact intersection reading is available. Keep circles on guides. Correct swapped centres/radii rather than merely labels. The triangle inequalities hold; do not drag A onto BC. These circles establish side lengths, not an incircle or circumcircle.

Derive four and six from a two-to-three ratio

AD requires BD:DC=AB:AC=6:9=2:3. Ten split into five shares gives two per share, so BD=4 and DC=6. Check 4+6=10 and place D four units right of B, at screen (840,1000). Do not copy lengths 6 and 9 onto a base of 10, or use lengths 2 and 3 without scaling. Preserve letter order: BD=6, DC=4 reverses the required ratio.

Join AD and mark equal angles, not equal base segments

Join A to D and use matching small angle arcs. Do not put equal-length ticks on base pieces 4 and 6. Equal angles and unequal base segments coexist. Move labels rather than points if crowded and soften reference circles. Marks state the intended construction, not a visual proof. Ratio plus the theorem’s converse supports D; disclose approximate drawing rather than treating stroke-width concealment as precision.

Use midpoint five as a separate control

Mark midpoint M separately: BM=MC=5, screen M=(900,1000). Draw AM as a pale dashed control, not merged with AD. Ratio 1:1 differs from 2:3, so AM is a median, not this bisector. This scalene triangle intentionally separates D and M; an isosceles special case changes no conclusion. Enlarge locally or move labels if points merge visually, without changing 4 or 5. Keep the control recoverable in the native work.

Audit ratio and total independently

Check 4/6=6/9, or cross-products 4×9=6×6=36, separately from 4+6=10. Equal total alone admits wrong 5+5; ratio alone still needs the declared total. Verify B,D,C collinear with D internal, not beyond C under an internal-division formula. Correct mistaken comparison pairs such as 4:9 explicitly. External arithmetic supports manually arranged geometry; it is not automatic app validation.

Troubleshoot swapped labels and left/right order

If D lands at the midpoint, check median confusion; if too near C, check side-label reversal. Convert radius to diameter once, not twice through scale. Restore the chosen upper intersection, or explicitly reflect the whole diagram with consistent letters, never one side alone. Inspect whether ordinary guide shapes export rather than assuming native non-exporting guides. After each correction recheck total, ratio and point order instead of moving points and labels together.

Reopen and explain where four and six come from

Save native work, export PNG and reopen. Read triangle/AD small and side lengths, base pieces and angle arcs enlarged. If M is exported, distinguish its dashed legend; if hidden, retain the native control. Reopen editable points/lines/labels and document actual approximate or numeric placement. Success derives 4 and 6 from ratio 2:3 and total 10 and rejects 5/5, without claiming universal base bisection, automatic angle measurement or completed precision testing.

Related tutorials and sources

VectorPaint triangle medians: check midpoints and the two-thirds centroid position; VectorPaint 3–4–5 triangles: check circumcircle vertices and incircle sides separately.

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 VI.3 gives the internal-bisector side ratio and converse. Values, layout and counterexample are original; reference circles and checking lines are ordinary manually arranged shapes.