Construct and verify a motion witness

Try

Suppose a triangle must be placed on a target without stretching. “They look alike” is not an executable instruction. A witness is an ordered list of motions, with enough information to apply each motion to every point. Labels matter: the required image of A might differ from the image that gives an unlabeled overlap.

As a separate worked example, take A=(0,0), B=(1,0), C=(0,2). Rotate 90 degrees counterclockwise about the origin and then translate by (3,1). The intermediate vertices are (0,0),(0,1),(−2,0); the target vertices are (3,1),(3,2),(1,1). This fixed example is independent of the live explorer’s starting values.

Predict

Predict the combined coordinate rule and the result of reversing those two steps. Then decide what evidence could prove that no rigid witness exists for a different target. A failed attempt is not an impossibility proof: perhaps the correct reflection or order has not been tried. A mismatched corresponding distance, however, rules out every rigid-motion composition at once.

Experiment

Build a motion witness

Map each labeled source vertex to the same label on the target. Build a sequence of three operations; leave unused steps as “No change.” Predict the intermediate coordinates before applying the sequence. Rotation is counterclockwise about the origin; reflection uses the selected line.

Target challengeMove the triangle to a new positionMatch a reversed orientationInvestigate a stretched targetReset motion sequence

Step 1

Operation 1No changeTranslateRotate about originReflectHorizontal stretch ×2 (non-rigid)

(x,y) → (x,y)

Step 2

Operation 2No changeTranslateRotate about originReflectHorizontal stretch ×2 (non-rigid)

(x,y) → (x,y)

Step 3

Operation 3No changeTranslateRotate about originReflectHorizontal stretch ×2 (non-rigid)

(x,y) → (x,y)

Apply motion sequence

Same labels specify correspondence

Vertex	Source	Target	Stage
A	(-2,-1)	(1,0)	Hidden
B	(0,-1)	(3,0)	Hidden
C	(-2,2)	(1,3)	Hidden

Dashed: source. Dotted: target. Solid with vertex dots: inspected stage. These style differences supplement color. Coordinates outside the fixed window are still retained and checked.

Results are hidden until you apply the sequence. Editing an operation clears the old result.

These affine maps take each straight edge to the edge between its mapped endpoints, so matching all three vertices matches this whole filled triangle. A few point matches do not prove an arbitrary transformation preserves every distance. The lesson’s general arguments establish why translations, rotations and reflections are rigid. Temporary exploration does not award mastery.

Choose a target in the explorer and write down a three-step plan before running it. Inspect the intermediate frames and the labeled final vertices. Record two separate decisions: do the vertices match, and are all the selected steps rigid? The available horizontal stretch is deliberately a diagnostic counterexample. Matching a target by that step would not prove rigid congruence.

For the orientation-reversed target, compare an attempted translation-and-rotation sequence with one containing a reflection. For the stretched target, inspect the exact squared-distance table before searching indefinitely. Approximate angle readouts help inspection, but the exact distance obstruction supplies the decisive argument. Points outside the fixed plotting window remain part of the mathematical figure; clipping is not deletion.

Understand

For the separate worked example, the quarter-turn maps (x,y) to (−y,x), then translation gives (3−y,1+x). Reversing the steps gives (−y−1,x+3), a different function. Both functions are rigid, yet only the first realizes the specified target correspondence. Checking A alone already detects the reversal: the outputs are (3,1) and (−1,3).

Vertex	After quarter-turn	After translation
A=(0,0)	(0,0)	(3,1)
B=(1,0)	(0,1)	(3,2)
C=(0,2)	(−2,0)	(1,1)

The source squared side lengths are 1, 4 and 5. The target has the same corresponding values. This is necessary for any rigid witness. Our explicit composition proves more: its invertible point rule maps the three edges and their enclosed region onto the target triangle. We have not merely placed three isolated points.

To see a useful obstruction, replace the target B by (3,4) while keeping its A at (3,1). The corresponding AB length would change from 1 to 3. Every rigid motion preserves that distance, and a composition preserves it step after step. Therefore no rigid witness can satisfy those labels. An exhaustive search through a few menu choices is not needed and would be weaker than this invariant argument.

Orientation supplies another distinction. Walking around a nondegenerate labeled triangle in order A,B,C can be clockwise or counterclockwise. Translations and rotations preserve that order; a reflection reverses it. A reversed target can still be congruent, but a sequence containing only translations and rotations cannot realize that labeled correspondence. Two reflections can restore orientation, so count the effect of the whole composition rather than treating the word “reflection” as a permanent obstacle.

Apply

Write a witness report with source coordinates, target coordinates, ordered operations, intermediate coordinates and a general reason the operations are rigid. If your result fails, identify the first step where the trace differs from your plan. If you claim impossibility, name an invariant that fails, such as a corresponding distance. “I could not find it” describes your search, not the geometry.

A construction may use motions outside the explorer’s finite menu; the written practice accepts precise coordinate or compass-and-straightedge descriptions. The bounded menu supports reasoning but does not define the set of all Euclidean motions.

Continue

The next lesson uses such witnesses to define triangle congruence. After that we derive criteria that guarantee a witness from a small set of side and angle data, instead of assuming those criteria from a picture.