VectorPaint two tangents from an external point: check right angles and equal lengths

English translation of the Chinese original, translated on 2026-11-14. Original publication dates, attribution, code and version notes are retained.

Draw two tangents from one external point. Looking as though a line touches a circle is insufficient: preserve the centre, contact points and perpendicularity evidence. Calculate on paper; VectorPaint organises paths and annotations without serving as an automatic tangent solver.

Fix the coordinates, scale and external point

Create a 1600×1200 landscape canvas and first record mathematical O=(0,0), radius 3 and P=(5,0). Use 100 canvas units per mathematical unit, with the displayed centre at (600,600). Mathematical y points up; screen y points down. The circle diameter is 600 and P is displayed at (1100,600). Draw the circle through available shape controls and retain a centre cross. Separate guides, tangents and text, declaring OP=5 greater than radius 3. Establish the external-point condition rather than estimating it from an enlarged visual gap.

Calculate contacts using the perpendicular-radius condition

Write T=(x,y). Being on the circle requires x²+y²=9; OT perpendicular to PT gives x(5−x)−y²=0. Combining these gives 5x=9. Both contacts therefore have x=1.8, with y=2.4 and −2.4. Consult Euclid’s Book III tangent construction for the problem of constructing a tangent from a given point. This coordinate calculation is our own audit, not a claim that the historical text uses these decimals or that the app has a matching calculator button.

Plot both contacts and connect radii and tangents

Place the upper contact at screen (780,360) and the lower at (780,840), labelling them T-upper and T-lower. Connect O to each before joining P to each. Endpoints must share the contact position, not a decorative dot beyond the circumference. Zoom during node editing to distinguish the paths: the circumference continues through the contact, while radius and tangent segment terminate there. Do not break the circle and leave a gap. Keep contact markers on a separate layer so they cannot conceal a bad join.

Check circle membership before perpendicularity

Substitute 1.8²+2.4²=3.24+5.76=9 for both contacts. From the upper contact to P the vector is (3.2,−2.4); its dot product with radius vector (1.8,2.4) is 5.76−5.76=0. At the lower contact, (3.2,2.4) dotted with (1.8,−2.4) also vanishes. Add right-angle markers only at those contacts. A steep-looking line does not establish perpendicularity, and checking one side does not constitute an actual device measurement of the other.

Separate tangent length four from centre distance five

Each tangent length is √(3.2²+2.4²)=4, independently confirmed by √(5²−3²). Put 4 on each PT segment, 5 on OP and 3 on each radius. Five is not the tangent length, and adding the two fours does not give the shortest distance from P to the circle. The nearest distance along OP is 2; identify it only in a side note distinguishing the quantities. For a boundary control replace the external point by the centre: two real tangents no longer apply because it is inside, and the external distance is no longer 5. Separate annotation leaders so letters, rather than colour alone, still identify each measurement in grayscale.

Make a secant control through the circle

Duplicate an unapproved control at the right. Draw a horizontal line from P toward the centre through the circle, marking its two intersections as a secant. Extending either valid tangent should still produce one circumference contact; do not bend it into a short arc hugging the circle merely for visibility. Name valid and invalid groups separately and distinguish straight lines from the circular curve. Fade and label any auxiliary circle or diameter so readers do not mistake auxiliary intersections for contacts of the two tangents.

Troubleshoot changes that repair only one side

After moving the centre or resizing the circle, check whether P and both contacts received the same uniform transformation. Enlarging only the circle invalidates the old tangents. Re-read coordinates and substitute into both conditions; a close-looking undo result is insufficient. If exact coordinate controls are unavailable, position manually against the grid and label the drawing approximate while retaining exact paper results. Inspect right-angle symbols and decimal points in output. Stroke thickness cannot prove tangency or establish motion, friction or manufacturing precision.

Reopen and audit both contacts and lengths

Save the editable document, leave it and reopen. Hide the answer text and recover radius 3, external distance 5 and both tangent lengths 4 from O, P and the contacts. Restore answers and check circle membership, both zero dot products and equal lengths, confirming that y reversal occurs only in display conversion. Open final output and inspect circular proportions, the secant control label and unobscured contacts. Success means another reader can rebuild and verify both tangents numerically; unexamined formats and device behaviour remain unverified.