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Interactive Geometry

Fixed Point of a Similarity

Start with the visual riddle: where is the hidden center that the whole motion seems to turn around? Then switch to explore mode to see the controls and the theorem behind it.

Fixed Point of a Similarity
Play it as a visual riddle first, then switch to explore mode when you want the controls and theorem view.
AA'PVisual riddle: find the hidden centerClick where the whole motion seems to pivot.

Your Goal

Guess the one point that stays fixed while the blue shape is rotated, scaled, and shifted.

Your Guess

No guess yet

Score

Place a guess to see how close you are

Riddle Mode

Like the swirl in a painting, this motion feels as if the whole figure is turning around a secret center. Your job is just to find that center by eye.

Click anywhere on the board to place your guess.

Quick setup

Shape

Theorem Engine
The algebra behind the visual stays separate from the UI.

Every similarity of the plane can be written as

T(x)=kRθ(x)+v.T(x) = kR_\theta(x) + v.

A fixed point PP satisfies

P=kRθ(P)+v,P = kR_\theta(P) + v,

so

(I−kRθ)P=v.\left(I - kR_\theta\right)P = v.

If det⁡(I−kRθ)≠0\det\left(I - kR_\theta\right) \neq 0, this point is unique. The only time the determinant vanishes for our positive-scale model is the pure translation case k=1k = 1 and θ=0∘\theta = 0^\circ.

What To Notice

Game mode is intentionally lighter: click a guess, check your error, and reveal the answer only if you want it.

Explore mode keeps the original controls, including the scale, rotation, translation, and the live angle and ratio readout for a chosen vertex.

Hit Translation Only to see the one case where the hidden center disappears.