VectorPaint 3–4–5 triangles: check circumcircle vertices and incircle sides separately

English translation of the Chinese original, translated on 2026-11-12. Original publication dates, attribution, code and version notes are retained.

Triangle centres need not coincide. Copy a 3–4–5 right triangle into two views: one circumcircle through its vertices and one incircle tangent to its sides. Check the different conditions separately, without an automatic triangle-centre tool or treating visual proximity as exact tangency.

Fix the triangle before copying it

Use A=(0,0), B=(4,0), C=(0,3), right angle at A, side lengths 4,3,5 and area 6. Draw once at a fixed scale, then translate a copy rather than making a more balanced second triangle. At 100 pixels per unit, subtract mathematical y for downward screen coordinates. TI’s centre lesson supplies perpendicular-bisector versus angle-bisector definitions, not calculator-installation steps for VectorPaint. Do not rely on its sentences that confuse circle and triangle locations.

Use the hypotenuse midpoint for the circumcentre

The midpoint of BC is O=(2,1.5); its distance to B is √(2²+1.5²)=2.5. OA, OB and OC all equal 2.5, so this circle passes through all vertices. O lies on BC, not inside this triangle; do not move it inward for label space. The perpendicular bisectors of the legs, x=2 and y=1.5, independently locate O. Hide ordinary drawn reference lines after checking. Move text for readability without moving its point.

Check vertices rather than sides for the circumcircle

Create an equal-width/height circle positioned by centre and diameter 5, explicitly approximate if dragged visually. Check A,B,C lie on it; tangency to all sides is not required. Much of the circle legitimately lies outside the triangle. A mask leaving only its interior arc is not a complete circumcircle. If B/C fit but A does not, inspect centre or accidental ellipticity rather than dragging A onto the circle and changing the 3–4–5 triangle.

Check incenter by perpendicular distances to sides

In the second view set I=(1,1), radius 1. Its distances to horizontal AB and vertical AC are both 1. BC has equation 3x+4y=12, giving distance (12−3−4)/5=1. Equal side distances, not visual centrality, establish the condition. Area 6 divided by semiperimeter 6 independently gives inradius 1. I and O are different objects; relabelling the first circle is insufficient. Coordinates refer to each view’s own origin, unaffected by layout translation.

Distinguish tangency points from vertices

Incircle contact points are (1,0) on AB, (0,1) on AC and (1.6,1.8) on BC. The last satisfies 3×1.6+4×1.8=12 and lies distance 1 from I. A radius to a contact point is perpendicular to its side; a line to a vertex is not that radius. IA=√2, not 1, so the incircle does not pass through A and need not pass through B/C. Reduced output may retain contact dots and one right-angle cue, while the complete checking copy preserves all correspondences. The incircle spans 0–2 on both axes; the circumcircle spans x=−0.5 to 4.5 and y=−1 to 4. Allow that space rather than flattening it into the triangle’s bounding box. White margins may differ while mathematical scale stays equal.

Swap conditions in an erroneous copy

In a wrong second-view copy, enlarge the I-centred circle to pass through A, making radius √2 and losing tangency to AB/AC. This swaps conditions rather than improving the incircle. Another error labels O as (1,1), whose distances to vertices are unequal. Keep separate copies and calculate distances without moving both triangle and circle, which would destroy the control. Restore centre/radius before rechecking contacts or vertices, not merely legend names. Colour proves neither circle type; monochrome labels and contacts must identify them.

Preserve common references in two-view output

Save and open a PNG, first checking identical side lengths, right-angle location and letters across both views, then circumcircle vertices and incircle sides separately. Move labels away from contacts without moving points. A distorted circle requires width/height or display checks, not altered mathematical radius. Hide ordinary drawn bisectors/radii as appropriate and inspect native references in actual output. Open SVG/PDF separately for line/text placement, while retaining numerical checks; one display establishes neither absolute precision nor identical rendering everywhere.

Accept separate three-vertex and three-side tests

Acceptance requires O=(2,1.5), R=2.5 with equal vertex distances; I=(1,1), r=1 with equal side distances; and all three contact points. If both circles are forced inside, revisit definitions. Mixed radii require distance checks; a misplaced sloping-side contact requires its equation. Retain area 6 and semiperimeter 6 without calling 6 the perimeter, which is 12. Other triangles require recalculation: circumcentres are not always on an edge. This is manual geometry, with no automatic centre-solving or physical calibration claim.