These may be joined, forming an equilateral triangle in which there are three vertices or points, three lines or sides and one surface.
At the vertices of a regular tetrahedron may be found such points.
Now the plane, which is tangent to the three spheres, is also evidently tangent to the three conical surfaces; and therefore the vertices of those conical surfaces are all in the tangent plane.
Distances can easily be measured by constructing a large equilateral triangle of heavy pasteboard, and standing pins at the vertices for the purpose of sighting.
Evidently three stays are needed, and they are preferably placed at the vertices of an equilateral triangle, the smokestack being in the center.
The point from which lines are drawn to the vertices is usually taken at a vertex, so that there are n - 2 triangles.
A cube may be inscribed in an octahedron, its vertices being at the centers of the faces of the octahedron.
The theorem asserts that in any convex polyhedron the number of edges increased by two is equal to the number of vertices increased by the number of faces.
At the vertices of this polygon the Pythagoreans placed the Greek letters signifying "health.
If we think of all the vertices projected on ABCDE, by lines through P, the sum of the angles of all the faces will be the same as the sum of the angles of all their projections on ABCDE.
The practical problem may be given of locating the vertices of the triangle and of finding the length of each stay.
Let us change the vertices into representations, and see what will be the result.
It is called quadrilateral, and, like the triangle, has its sides and vertices of angles.
It is called a triangle; the lines are called sides, and the points where they unite the vertices of the angles.
There exists an infinity of surfaces of the second degree, which at one of their vertices osculate any surface whatever at a given point.
Thus, The three pairs of rays which may be drawn from a point through the three pairs of opposite vertices of a complete quadrilateral are said to be in involution.
The four lines joining the two opposite pairs of vertices and the two opposite points of contact of a quadrilateral circumscribed about a conic all meet in a point.
The three tangents at the vertices of a triangle inscribed in a conic meet the opposite sides in three points on a straight line.
For the circumscribed quadrilateral, Brianchon’s theorem gave (§ 88) The lines joining opposite vertices and the lines joining opposite points of contact are four lines meeting in a point.
If a quadrilateral be circumscribed about a conic, the lines joining two pairs of opposite vertices and the lines joining two opposite points of contact are four lines which meet in a point.
We derived as a consequence of the theorem of Brianchon (§ 89) the proposition that if a triangle be circumscribed about a conic, the lines joining the vertices to the points of contact of the opposite sides all meet in a point.
Two pairs of vertices may coalesce, giving an inscribed quadrangle.
The lines joining the vertices to the points of contact of the opposite sides of a triangle circumscribed about a conic all meet in a point.
If the six points be called the vertices of a hexagon inscribed in the curve, then the sides 12 and 45 may be appropriately called a pair of opposite sides.
Finally, three of the vertices of the hexagon may coalesce, giving a triangle inscribed in a conic.
But these four points form a quadrangle inscribed in the conic, and we know by § 95 that the tangents at the opposite vertices γ and γ’ meet on the line v.
The head is subcordate, with the vertices rounded.
The head is slightly bifid, with the vertices rounded.
It is further of interest to note that if the weights be all equal, and at equal horizontal intervals, the vertices of the funicular will lie on a parabola whose axis is vertical.
The points thus obtained are evidently the vertices of a polyhedron with plane faces.
The force-diagram is constructed by placing end to end a series of vectors representing the given forces in magnitude and direction, and joining the vertices of the polygon thus formed to an arbitrary pole O.
If in fact we take the pole of each face of such a polyhedron with respect to a paraboloid of revolution, these poles will be the vertices of a second polyhedron whose edges are the "conjugate lines" of those of the former.
If we take any polyhedron with plane faces, the null-planes of its vertices with respect to a given wrench will form another polyhedron, and the edges of the latter will be conjugate (in the above sense) to those of the former.
The two focal lengths and the distances of the foci from the verticesbeing known, the positions of the remaining cardinal points, i.
We proceed to determine the distances of the focal points from the vertices of the lens, i.
FHH'F'; and the relation of the principal points to the vertices is also the same as in the mensicus.
In 1743 Thomas Simpson in his Mathematical Dissertations published a very convenient rule, obtained by taking the vertices of three consecutive equidistant ordinates to be points on the same parabola.
Wrought iron has in itself certain parts Boreal & Austral: a magnetick vigour, verticity, and determinate vertices or poles.
The ends of these diameters are the vertices of the required square.
If a be inscribed in a curve of the hexagon be circumscribed about second order, then the a curve of the second class, then intersectionsof opposite sides the lines joining opposite vertices are three points in a line.
This side being known the decagon can be constructed, and if the vertices are joined alternately, leaving out half their number, we obtain the regular pentagon.
Hence P and Q are two of the vertices on the polar-triangle which is determined by the four-point ABCD.
The harmonic points are the vertices of the "harmonic triangle of the complete quadrangle.
A polygon is said to be inscribed in a circle, and the circle is said to be circumscribed about the polygon, if the vertices of the polygon lie in the circumference of the circle.
If, therefore, one side cuts a conic, then one of the twovertices which lie on this side is within and the other without the conic.
Of a polar-triangle any two vertices are conjugate poles, any two sides conjugate lines.
If the sides of one triangle meet those of another in three points which lie in a line, then the vertices lie on three lines which meet in a point.
In order, therefore, that a spheroidal triangle may be exactly defined, it is necessary that the nature of the lines joining the three vertices be stated.
Or again-- The projections from any point on to any line of the six vertices of a four-side are six points in involution, the projections of oppositevertices being conjugate points.
The above list will hopefully give you a few useful examples demonstrating the appropriate usage of "vertices" in a variety of sentences. We hope that you will now be able to make sentences using this word.
