Every tool here is a proof you can step through on a live figure: each step names its justification, and you can drag the points at any step to see the argument hold for every case.
💡Tip: Start with the Triangle Angle Sum proof: press Play, then drag a vertex.
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Proofs
Proofs You Can Step Through
A proof is a chain of steps, each one justified by something already known: a postulate, a definition or an earlier theorem. The proof player shows that chain one step at a time on a live figure. Each step draws what it constructs, highlights the parts it uses and names the reason it is allowed. Drag a point at any step and the construction, the highlighted angles and the conclusion all follow the new figure. A proof is a claim about every figure of its kind, and dragging lets you see that the argument never depended on the one drawn on screen.
How to read a proof. Most geometric proofs follow the same order, and the player keeps it: Given, the figure and what is assumed about it; To prove, the claim, stated before any work starts; Construction, extra points, lines or circles added to make the argument possible; Argument, steps that each follow from the given facts, the construction or an earlier step; Conclusion, the claim again, now established, closed with the end-of-proof mark ∎. A construction is not part of the claim. It is a tool the proof adds, and the result holds whether or not the extra lines are drawn.
The justification tags. Every step in the log carries a small tag above its text. The tag names the rule that makes the step valid, so the reasoning can be checked one link at a time. In the triangle angle sum proof the tags are Given, Parallel postulate, Alternate angles and Straight angle. Each one is a fact that is either assumed or proved earlier, and none of them is a measurement. If a step has no rule that justifies it, it is not part of a proof. Reading the tags is the quickest way to see exactly what a theorem depends on, for example which results need the parallel postulate and which do not.
Dragging is not proving. Checking many triangles never proves a statement about all of them; there are always more triangles than any number of checks. The proof is the chain of justified steps. Dragging shows something different: that no step relied on a particular size or shape. Some positions are refused. If a drag would put the three vertices on one line, or nearly so, the figure stops being a triangle and the move is rejected, because the theorem says nothing about that case.
Triangle proofs. Triangle proofs come first because so much of plane geometry rests on them. The triangle angle sum proof shows that the interior angles of every triangle add up to 180°, using one parallel line and two pairs of alternate angles. Results about exterior angles, right triangles and polygon angle sums all follow from it.
Using the controls. Every proof page uses the same player. ▶ Play runs the proof at the chosen speed; Next → and ← Back move one step at a time. Reset returns to the first step and restores the original figure. Labels shows or hides the point names; Measures writes each angle's size beside its arc. The Step explanations log keeps every step reached so far; click an entry to return to it. Solid points can be dragged; hollow points are constructed from the others and follow them.