THE FOURTH BOOK.
The fourth Book of the Elements consists entirely of problems. The first five propositions relate to triangles of any kind; the remaining propositions relate to polygons which have all their sides equal and all their angles equal. A polygon which has all its sides equal and all its angles equal is called a regular polygon.
IV. 4, By a process similar to that in IV. 4 we can describe a circle which shall touch one side of a triangle and the other two sides produced. Suppose, for example, that we wish to describe a circle which shall touch the side BC, and the sides AB and AC produced: bisect the angle between AB produced and BC, and bisect the angle between AC produced and BC; then the point at which the bisecting straight lines meet will be the centre of the required circle. The demonstration will be similar to that in IV. 4.
A circle which touches one side of a triangle and the other two sides produced, is called an escribed circle of the triangle.
We can also describe a triangle equiangular to a given triangle, and such that one of its sides and the other two sides suppose AK produced to meet the circle again; and at the point of intersection draw a straight line touching the circle; this straight line with parts of NB and NC, will form a triangle, which will be equiangular to the triangle MLN, and therefore equiangular to the triangle EDF and one of the sides of this triangle, and the other two sides produced, will touch the given circle. the part which shews that DF and EF will meet. It has also been proposed to shew this in the following way: join DE; then the angles EDF and DEF are together less than the angles ADF and AEF, that is, they are together less than two right angles; and therefore DF and EF will meet, by Axiom 12. This assumes that ADE and AED are acute angles; it may however be easily shewn that BE is parallel to BC, so that the triangle ADE is equiangular to the triangle ABC and we must therefore select the two sides AB and AC such that ABC and ACB may be acute angles.
IV. 10. The vertical angle of the triangle in IV. 10 is easily seen to be the fifth part of two right angles; and as it
