Drag the points and watch each theorem hold, live. A collection of 73 hands-on geometry explorers — every construction verified to machine precision.
Drag a triangle or type exact side lengths, and watch a² + b² compare to c² live.
Drag points on a circle and watch the central angle stay exactly twice the inscribed angle on the same arc.
Generalize the Pythagorean theorem to any triangle.
Drag any polygon and watch its interior angles always add up to (n−2)×180°.
Walk around any polygon — the turns you make always add up to a full 360° circle.
Type three side lengths and see exactly when they can — and can't — close into a triangle.
Drag a triangle and watch the bisector split the opposite side in the same ratio as the two sides meeting at that corner.
Drag a point anywhere around a rectangle and see why its squared distances to opposite corners always stay balanced.
Drag a point inside an equilateral triangle and watch its three distances to the sides always total the same height.
Drag two lines across a circle and see why the products of their segment lengths from the crossing point always match.
Drag a quadrilateral inscribed in a circle and see why its diagonals' product equals the sum of its opposite sides' products.
Drag three cevians through a shared point and see why the side-division ratios always multiply to exactly one.
Drag a triangle and watch nine special points — midpoints, altitude feet, and more — all land on one circle.
Drag any quadrilateral and watch the midpoints of its sides always form a perfect parallelogram.
Drag a line across a triangle and see why the three side-division ratios it makes always multiply to one.
Drag a tangent and chord on a circle and see the angle they make equal the inscribed angle in the far arc.
Drag two secants from an outside point and see the angle equal half the difference of the arcs they cut.
Drag a triangle and watch its three medians meet at the centroid, splitting each in a 2:1 ratio.
Slide a line parallel to a triangle's base and see it split the other two sides in equal ratios.
Build equilateral triangles on each side of any triangle — their centers always form an equilateral triangle.
Trisect a triangle's angles and watch the adjacent trisectors meet at a perfect equilateral triangle.
Reshape a quadrilateral around an inscribed circle and see why its opposite sides always sum equally.
Drag a triangle and watch its circumcenter, centroid, and orthocenter stay on one line in a 1:2 ratio.
Slide a point around a triangle's circumcircle and watch the three perpendicular feet stay in a line.
Reshape a quadrilateral inscribed in a circle and see its area come straight from the four side lengths.
Type or drag a triangle's three sides and watch its area appear straight from the side lengths.
Drag a triangle and watch its angle bisectors meet at the incenter, the center of the inscribed circle.
Drag a triangle and watch the perpendicular bisectors meet at the circumcenter, equidistant from all vertices.
Drag a triangle and watch its three altitudes always pass through one point.
Join the midpoints of a triangle's sides and see each segment run parallel to a side at half its length.
Extend a triangle's side and see the exterior angle equal the sum of the two remote interior angles.
Slant a line across two parallel lines and explore corresponding, alternate, and co-interior angles.
Drag a triangle and watch its three interior angles always add up to 180°.
Cross two lines and watch the opposite angles stay equal while neighbours add to 180°.
Make two sides equal and watch the base angles opposite them stay equal too.
Drag a point and see it stay equally far from two fixed points exactly when it's on the bisector.
Drag a point inside an angle and see it stay equally far from both arms exactly on the bisector.
Slide a shared ray and see two angles always total 90° or 180°.
Drag the touch point around a circle and see the tangent always meet the radius at a right angle.
Drag a point outside a circle and see its two tangent segments stay equal in length.
Spin rays from a point and watch all the angles around it always total 360°.
Drag two points on a grid and read off their distance and midpoint from the coordinates.
Flip a triangle across a mirror line and see the image stay congruent with orientation reversed.
Spin a triangle about a center and see distances kept and orientation preserved.
Slide a triangle by a vector and see every point shift the same way, congruent and un-turned.
Scale a triangle from a center and see it grow or shrink while staying the same shape.
Sweep a central angle and see the arc and sector as a fraction of the whole circle.
Drag a polygon's corners on a grid and get its area straight from the coordinates.
Change the number of sides and see Area = ½·apothem·perimeter approach a circle.
Two equal angles make triangles similar — scale a second triangle and watch every side ratio stay equal.
Drag four corners around a circle and watch each pair of opposite angles always sum to 180°.
Slide a cevian's foot along a triangle's side and watch b²m + c²n = a(d² + mn) hold.
The altitude to a right triangle's hypotenuse gives h²=pq and each leg²=segment·hypotenuse.
Cross two chords inside a circle — the angle equals half the sum of the arcs they cut off.
Skew a parallelogram and watch its diagonals² always equal twice the sum of its sides².
Count grid dots inside and on a lattice polygon: its area is I + B/2 − 1.
Two sides and their included angle give the area directly: ½·a·b·sin C.
An inscribed circle splits a triangle into three pieces of height r, so its area is r·s.
The perpendicular from a circle's center always bisects the chord: AM = MB.
The five tip angles of any pentagram always add up to exactly 180°.
The segment joining the legs' midpoints equals the average of the two parallel sides.
A parallelogram's diagonals always cross at their shared midpoint: AM = MC, BM = MD.
Each side over the sine of its opposite angle equals the circumscribed circle's diameter, 2R.
All four sides equal forces the diagonals to cross at right angles and bisect each other.
Two chords are the same length exactly when they sit the same distance from the center.
Reflecting across two mirrors that meet equals one rotation by twice the angle between them.
A triangle's area is the product of its sides over four times its circumradius.
In a right triangle, the median from the right angle is exactly half the hypotenuse.
Reflect a triangle across a line, then slide it along that line — the flipped image whose P→P″ midpoints all land on the mirror.
Make two sides proportional with equal included angles and watch the triangles become similar — every side ratio matching.
Drop a perpendicular from the circumcenter to each side — their signed lengths always add up to the circumradius plus the inradius.
Draw the two tangents from a point outside a circle — the angle between them plus the central angle always makes a straight 180°.
The distance between a triangle's two centers gives OI² = R² − 2Rr, forcing the circumradius to be at least twice the inradius.