Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/141

Rh Wherefore the triangle LMN is equiangular to the triangle DEF, and it is described about the circle ABC.

PROPOSITION 4. PROBLEM.

To inscribe a circle in a given triangle.

Let ABC be the given triangle: it is required to inscribe a circle in the triangle ABC

Bisect the angles ABC, ACB, by the straight lines BD, CD meeting one another at the point D; [I.9 and from D draw DE, DF, DG perpendiculars to AB, BC, CA. [1.12. Then, because the angle EBD IS equal to the angle FBD, for the angle ABC is bisected by BD [Construction. and that the right angle BED is equal to the right angle BFD; [Axiom 11. therefore the two triangles EBD, FBD have two angles and the side BD, which is opposite to one of the equal angles in each, is common to both; therefore DE is equal to DF.

For the same reason DG is equal to DF. Therefore DE is equal to DG. [Axiom 1. Therefore the three straight lines DE, DF, DG are equal to one another, and the circle described from the centre D, at the distance of any one of them, will pass through the extremities of the other two; and it will touch the straight lines AB, BC, CA because the angles at the points E, F, G are right angles, and the straight line which is drawn from the extremity of a dia- meter at right angles to it, touches the circle. [III. 16. Cor. Therefore the straight line AB,BC,CA do each of them touch the circle, and therefor the circle is inscribed in the triangle ABC.

Wherefore a circle has been inscribed in the given triangle.