Curiosa Mathematica, Part 1

Such a Pair of Lines may be said to be 'equally inclined' to that transversal.

If two isosceles Triangles have equal bases but unequal sides: that Triangle, which has the greater sides, has the greater area.

Let the Triangles be set on the same base, and call them , . And let , , be respectively greater than , .

Now cannot fall within the Triangle , or upon either of its sides; for then , would be less than , ;

neither can it fall on ; for then , would be equal to , ;

neither can intersect , nor intersect ; for, in either case, if were joined, there would be two Triangles, on the same base , having their coterminous sides equal; which is impossible;

Therefore, if two isosceles Triangles &c.

Given a certain angle; and given that any isosceles Triangle, whose vertical angle is not-greater than the given angle, has its base not-greater than either of its sides: to describe, on a given base, an isosceles Triangle having each base-angle equal to the given angle.

At , in Line , make angle equal to given angle, making ; and join .

Then, by hypothesis, is not-greater than : i. e. it is either equal to it, or less than it.

Then Triangle is equilateral: i. e. it is an isosceles Triangle, on given base , and having each base-angle equal to given angle.

Then Triangle is isosceles, and is on given base , and has each base-angle equal to given angle.

The isosceles Triangle, so described, has its vertical angle not-less than either of its base-angles.