Angle bisector
An arc cuts both arms of an angle, equal arcs from those cuts meet, and the ray drawn through the meeting point halves the angle exactly.
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Euclid
15 animationsAngle in a semicircle
The vertex is taken to four points of the arc in turn and the angle it subtends on the diameter stays a right angle at every one of them.
CSSChords of a circle
Two chords are swung about a circle at different rates and the perpendicular dropped from the centre to each one lands on its midpoint however far it turns.
CSSCircumcircle
The perpendicular bisectors of the three sides are struck in turn and the circle drawn on the point where they meet passes through all three vertices.
CSSCongruent triangles
Two sides and the angle between them are marked on both figures, and one triangle is carried over and set down on the other so every marked part registers.
CSSEllipse by its foci
A point runs once round an ellipse while two bars read off its distances to the foci, one lengthening exactly as far as the other shortens.
CSSEquilateral triangle
Two compass circles of equal radius are struck from the ends of a segment and the triangle closes through the point where they cross.
CSSFocus and directrix
A circle drawn on a point travelling the parabola always sits on the focus and just touches the directrix, which is the definition of the curve.
CSSGeometric mean
The altitude to the hypotenuse cuts a right triangle into two that are each similar to the whole, the accent moving between the halves.
CSSGolden section
An arc from the midpoint of a square base through the far corner meets the base extended at the golden section, closing a rectangle in mean and extreme ratio.
CSSInscribed hexagon
The compass is opened to the radius and walked around the circle, each step marking the next vertex until six sides close the figure exactly.
CSSPerpendicular bisector
Equal arcs from both ends of a segment cross above and below it, and the line through the crossings cuts it in half at a right angle.
CSSPythagorean relation
A square is raised on each side of a right triangle and the two on the legs hand their accent across to the square on the hypotenuse.
CSSSimilar triangles
A triangle rides out along three rays from a fixed centre until it lands on a larger outline, its angles unchanged and its sides in constant ratio.
CSSTangent and radius
A tangent is rolled right around a circle and the radius drawn to the point of contact stays square to it the whole way round.
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