Statement
The axiom states that, in the plane,
- If a line intersects one side of a triangle internally then it intersects precisely one other side internally and the third side externally, if it does not pass through a vertex of the triangle.
A more informal version of the axiom is often seen:
- If a line, not passing through any vertex of a triangle, meets one side of the triangle then it meets another side.
In this statement, a side of a triangle is taken to be the line segment joining two vertices of the triangle, so the expression "a line meeting a side" means that the line meets the line forming the side of the triangle internally.
Read more about this topic: Pasch's Axiom
Famous quotes containing the word statement:
“Eloquence must be grounded on the plainest narrative. Afterwards, it may warm itself until it exhales symbols of every kind and color, speaks only through the most poetic forms; but first and last, it must still be at bottom a biblical statement of fact.”
—Ralph Waldo Emerson (1803–1882)
“If we do take statements to be the primary bearers of truth, there seems to be a very simple answer to the question, what is it for them to be true: for a statement to be true is for things to be as they are stated to be.”
—J.L. (John Langshaw)
“One is apt to be discouraged by the frequency with which Mr. Hardy has persuaded himself that a macabre subject is a poem in itself; that, if there be enough of death and the tomb in one’s theme, it needs no translation into art, the bold statement of it being sufficient.”
—Rebecca West (1892–1983)