GeoProof

A collection of 73 hands-on geometry explorers. Drag the figure and watch each classic theorem hold, live — every construction verified to machine precision. Pick a topic below, search by name, or test yourself with the practice quiz or a guided path. Everything is free, and nothing needs to be installed.

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Triangles

• Pythagorean Theorem — Drag a triangle or type exact side lengths, and watch a² + b² compare to c² live.
• Law of Cosines — Generalize the Pythagorean theorem to any triangle.
• Triangle Inequality — Type three side lengths and see exactly when they can — and can't — close into a triangle.
• Angle Bisector Theorem — Drag a triangle and watch the bisector split the opposite side in the same ratio as the two sides meeting at that corner.
• Viviani's Theorem — Drag a point inside an equilateral triangle and watch its three distances to the sides always total the same height.
• Ceva's Theorem — Drag three cevians through a shared point and see why the side-division ratios always multiply to exactly one.
• Nine-Point Circle — Drag a triangle and watch nine special points — midpoints, altitude feet, and more — all land on one circle.
• Menelaus' Theorem — Drag a line across a triangle and see why the three side-division ratios it makes always multiply to one.
• Centroid & the 2:1 Median — Drag a triangle and watch its three medians meet at the centroid, splitting each in a 2:1 ratio.
• Basic Proportionality — Slide a line parallel to a triangle's base and see it split the other two sides in equal ratios.
• Napoleon's Theorem — Build equilateral triangles on each side of any triangle — their centers always form an equilateral triangle.
• Morley's Trisector — Trisect a triangle's angles and watch the adjacent trisectors meet at a perfect equilateral triangle.
• Euler Line — Drag a triangle and watch its circumcenter, centroid, and orthocenter stay on one line in a 1:2 ratio.
• Simson Line — Slide a point around a triangle's circumcircle and watch the three perpendicular feet stay in a line.
• Heron's Formula — Type or drag a triangle's three sides and watch its area appear straight from the side lengths.
• Incenter & Incircle — Drag a triangle and watch its angle bisectors meet at the incenter, the center of the inscribed circle.
• Circumcenter & Circumcircle — Drag a triangle and watch the perpendicular bisectors meet at the circumcenter, equidistant from all vertices.
• Orthocenter — Drag a triangle and watch its three altitudes always pass through one point.
• Triangle Midsegment — Join the midpoints of a triangle's sides and see each segment run parallel to a side at half its length.
• Exterior Angle Theorem — Extend a triangle's side and see the exterior angle equal the sum of the two remote interior angles.
• Triangle Angle Sum — Drag a triangle and watch its three interior angles always add up to 180°.
• Isosceles Triangle — Make two sides equal and watch the base angles opposite them stay equal too.
• Perpendicular Bisector — Drag a point and see it stay equally far from two fixed points exactly when it's on the bisector.
• Distance & Midpoint — Drag two points on a grid and read off their distance and midpoint from the coordinates.
• AA Similarity — Two equal angles make triangles similar — scale a second triangle and watch every side ratio stay equal.
• Stewart's Theorem — Slide a cevian's foot along a triangle's side and watch b²m + c²n = a(d² + mn) hold.
• Geometric Mean — The altitude to a right triangle's hypotenuse gives h²=pq and each leg²=segment·hypotenuse.
• Triangle Area (sine) — Two sides and their included angle give the area directly: ½·a·b·sin C.
• Area = r·s — An inscribed circle splits a triangle into three pieces of height r, so its area is r·s.
• Sine Rule & 2R — Each side over the sine of its opposite angle equals the circumscribed circle's diameter, 2R.
• Area = abc/4R — A triangle's area is the product of its sides over four times its circumradius.
• Median to Hypotenuse — In a right triangle, the median from the right angle is exactly half the hypotenuse.
• SAS Similarity — Make two sides proportional with equal included angles and watch the triangles become similar — every side ratio matching.
• Carnot's Theorem — Drop a perpendicular from the circumcenter to each side — their signed lengths always add up to the circumradius plus the inradius.
• Euler's Inequality — The distance between a triangle's two centers gives OI² = R² − 2Rr, forcing the circumradius to be at least twice the inradius.

Circles

• Inscribed Angle Theorem — Drag points on a circle and watch the central angle stay exactly twice the inscribed angle on the same arc.
• Power of a Point — Drag two lines across a circle and see why the products of their segment lengths from the crossing point always match.
• Ptolemy's Theorem — Drag a quadrilateral inscribed in a circle and see why its diagonals' product equals the sum of its opposite sides' products.
• Tangent–Chord Angle — Drag a tangent and chord on a circle and see the angle they make equal the inscribed angle in the far arc.
• Intersecting Secants Angle — Drag two secants from an outside point and see the angle equal half the difference of the arcs they cut.
• Pitot's Theorem — Reshape a quadrilateral around an inscribed circle and see why its opposite sides always sum equally.
• Brahmagupta's Formula — Reshape a quadrilateral inscribed in a circle and see its area come straight from the four side lengths.
• Tangent ⟂ Radius — Drag the touch point around a circle and see the tangent always meet the radius at a right angle.
• Equal Tangents — Drag a point outside a circle and see its two tangent segments stay equal in length.
• Arc Length & Sector Area — Sweep a central angle and see the arc and sector as a fraction of the whole circle.
• Cyclic Quadrilateral — Drag four corners around a circle and watch each pair of opposite angles always sum to 180°.
• Intersecting Chords Angle — Cross two chords inside a circle — the angle equals half the sum of the arcs they cut off.
• Chord & Center — The perpendicular from a circle's center always bisects the chord: AM = MB.
• Equal Chords — Two chords are the same length exactly when they sit the same distance from the center.
• Tangent–Tangent Angle — Draw the two tangents from a point outside a circle — the angle between them plus the central angle always makes a straight 180°.

Angles & Lines

• Angle Sum of Polygons — Drag any polygon and watch its interior angles always add up to (n−2)×180°.
• Sum of Exterior Angles — Walk around any polygon — the turns you make always add up to a full 360° circle.
• Parallel Lines & Transversal — Slant a line across two parallel lines and explore corresponding, alternate, and co-interior angles.
• Vertical Angles — Cross two lines and watch the opposite angles stay equal while neighbours add to 180°.
• Angle Bisector (Locus) — Drag a point inside an angle and see it stay equally far from both arms exactly on the bisector.
• Complementary & Supplementary — Slide a shared ray and see two angles always total 90° or 180°.
• Angles Around a Point — Spin rays from a point and watch all the angles around it always total 360°.
• Star Angle Sum — The five tip angles of any pentagram always add up to exactly 180°.

Quadrilaterals

• British Flag Theorem — Drag a point anywhere around a rectangle and see why its squared distances to opposite corners always stay balanced.
• Varignon's Theorem — Drag any quadrilateral and watch the midpoints of its sides always form a perfect parallelogram.
• Parallelogram Law — Skew a parallelogram and watch its diagonals² always equal twice the sum of its sides².
• Trapezoid Midsegment — The segment joining the legs' midpoints equals the average of the two parallel sides.
• Diagonals Bisect — A parallelogram's diagonals always cross at their shared midpoint: AM = MC, BM = MD.
• Rhombus Diagonals — All four sides equal forces the diagonals to cross at right angles and bisect each other.

Polygons & Area

• Shoelace Formula — Drag a polygon's corners on a grid and get its area straight from the coordinates.
• Regular Polygon Area — Change the number of sides and see Area = ½·apothem·perimeter approach a circle.
• Pick's Theorem — Count grid dots inside and on a lattice polygon: its area is I + B/2 − 1.

Transformations

• Reflection — Flip a triangle across a mirror line and see the image stay congruent with orientation reversed.
• Rotation — Spin a triangle about a center and see distances kept and orientation preserved.
• Translation — Slide a triangle by a vector and see every point shift the same way, congruent and un-turned.
• Dilation (Scaling) — Scale a triangle from a center and see it grow or shrink while staying the same shape.
• Two Reflections — Reflecting across two mirrors that meet equals one rotation by twice the angle between them.
• Glide Reflection — Reflect a triangle across a line, then slide it along that line — the flipped image whose P→P″ midpoints all land on the mirror.