Proposition I.21

If on one of the sides of a triangle, from its extremities, there be constructed two straight lines meeting within the triangle,1 the straight lines so constructed 2 will be less than the remaining two sides of the triangle, but will contain a greater angle.

On BC, one of the sides of the triangle ABC, from its extremities B, C, let the two straight lines BD, DC be constructed meeting within the triangle;

I say that BD, DC are less than the remaining two sides of the triangle BA, AC, but contain an angle BDC greater than the angle BAC.

For let BD be drawn through to E.

Then, since in any triangle two sides are greater than the remaining one, I.20

  • therefore, in the triangle ABE, the two sides AB, AE are greater than BE.

Let EC be added to each;

  • therefore BA, AC are greater than BE, EC.

Again, since, in the triangle CED,

  • the two sides CE, ED are greater than CD, let DB be added to each; therefore CE, EB are greater than CD, DB.

But BA, AC were proved greater than BE, EC;

  • therefore BA, AC are much greater than BD, DC.

Again, since in any triangle the exterior angle is greater than the interior and opposite angle, I.16 therefore, in the triangle CDE,

  • the exterior angle BDC is greater than the angle CED.

For the same reason, moreover, in the triangle ABE also,

  • the exterior angle CEB is greater than the angle BAC.

But the angle BDC was proved greater than the angle CEB;

  • therefore the angle BDC is much greater than the angle BAC.

Therefore etc.

  • Q. E. D.

References

Footnotes

  1. be constructed...meeting within the triangle.
    The word meeting is not in the Greek, where the words are ἐντὸς συσταθῶσιν. συνίστασθαι is the word used of constructing two straight lines to a point (cf. I. 7) or so as to form a triangle; but it is necessary in English to indicate that they meet. ↩

  2. the straight lines so constructed.
    Observe the elegant brevity of the Greek αἱ συσταθεῖσαι. ↩