This proposition proves the triangle inequality, stating that the sum of any two sides of a triangle is always larger than the remaining side. To prove this, Euclid leverages the previous proposition by combining AC and AB into one long line segment, CD. Specifically, AC is extended to D so that AD is of equal length with BC. Triangle ABD is isosceles, so the angle ABD equals the angle ADB. The angle CBD is obviously larger than the angle ABD, so it must also be larger than the angle ADB. This sets us up to use the previous proposition, proving that the length of CD is larger than the length of BC, in turn completing the proof.