— GEOMETOR • elements

G.33 right angle and perpendicular

When a segment standing on a line makes the adjacent angles equal, each of the equal angles is a right angle, and the standing segment is perpendicular to the line.

G.35 construct a perpendicular from a point to a line

Construct a perpendicular to a given line from a given point not on it.

G.94 law of cosines for obtuse triangles

In obtuse-angled triangles, the square on the side subtending the obtuse angle is greater than the sum of the squares on the sides containing the obtuse angle by twice the rectangle contained by one of those sides and the projection of the other upon it.

G.95 law of cosines for acute triangles

In acute-angled triangles, the square on the side subtending an acute angle is less than the sum of the squares on the sides containing the acute angle by twice the rectangle contained by one of those sides and the projection of the other upon it.

G.113 chords equally distant from the center

In a:term:circle <G.10>segments are said to be:term:**equally distant from the center** <equally-distant-from-center> when the:term:perpendiculars <G.33> drawn to them from the:term:center <center> are equal.

G.114 equal chords are equidistant from the center

In a circle, chords are equal if and only if they are equally distant from the center.

G.115 chord at greater distance from the center

And that:term:segment <G.4> is said to be:term:**at a greater distance** <greater-distance-from-center> on which the greater:term:perpendicular <G.33> falls.

G.116 diameter is the greatest chord

Of chords in a circle, the diameter is the greatest, and a chord nearer the center is always greater than one more remote.

G.120 radius to point of contact is perpendicular to tangent

If a line touches a circle, the segment joined from the center to the point of contact is perpendicular to the tangent.

G.143 intersecting chords theorem

If two chords in a circle intersect one another, the rectangle contained by the segments of one equals the rectangle contained by the segments of the other.

G.144 tangent-secant theorem

If from a point outside a circle a secant and a tangent are drawn, the rectangle contained by the whole secant and its external segment equals the square on the tangent.

G.154 inscribe a circle in a triangle

Inscribe a circle in a given triangle.

G.164 circumscribe a regular pentagon about a circle

Circumscribe an equilateral and equiangular pentagon about a given circle.

G.209 height of a figure

The height of any:term:figure <G.9> is the:term:perpendicular <G.33> drawn from the vertex to the base.

G.211 segment drawn parallel

If a segment be drawn parallel to one of the sides of a triangle, it will cut the sides of the triangle proportionally; and, if the sides of the triangle be cut proportionally, the line joining the points of section is parallel to the remaining side of the triangle.

G.218 right angled triangle

If in a right-angled triangle a perpendicular be drawn from the right angle to the base, the triangles adjoining the perpendicular are similar both to the whole and to one another.

G.513 elevated point draw

From a given elevated point to draw a segment perpendicular to a given plane.

G.518 segments meeting parallel

If two segments meeting one another be parallel to two segments meeting one another, not being in the same plane, the planes through them are parallel.

G.532 segment point construct

On a given segment, and at a given point on it, to construct a solid angle equal to a given solid angle.

G.545 there equal plane

If there be two equal plane angles, and on their vertices there be set up elevated segments containing equal angles with the original segments respectively, if on the elevated segments points be taken at random and perpendiculars be drawn from them to the planes in which the original angles are, and if from the points so arising in the planes segments be joined to the vertices of the original angles, they will contain, with the elevated segments, equal angles.

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