# VectorPaint 3–4–5 triangles: check circumcircle vertices and incircle sides separately
Language: en
Canonical: [VectorPaint 3–4–5 triangles: check circumcircle vertices and incircle sides separately]()
中文：[中文 HTML]()
English: [English HTML]()
Published: 2026-11-12T00:00:00+08:00
Modified: 2026-11-12T09:43:29+08:00
Author: Bison
Draw both centres for one triangle, checking circumradius 2.5, inradius 1 and tangency locations.
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.
## Related tutorials and sources
[VectorPaint ellipse-focus diagrams: distinguish semiaxes and the two focal distances](), [VectorPaint perpendicular bisectors: check equal-radius intersections and midpoint]().
Checked on November 12, 2026: VectorPaint public version 1.0.8, requiring iOS / iPadOS 17.0 or later. Scope follows the [official App Store]() and [Apple public lookup](). Manual exercises use controls actually available in your installed version; external calculations, written records and diagrams imply no built-in automatic analysis. No step-by-step physical-device, private-file or all-export-environment testing is claimed.