The figures illustrate the construction and proof.

Our proof, after that of Euclid, is based on copying one of the triangles and then showing that the other triangle is congruent to this copy. If the three sides of one triangle are equal to the three sides of another triangle, the triangles are congruent. We now move to a proof from one of the best modern high school geometry texts - Geometry: Seeing, Doing, Understanding by H. The points A and A 1 can't therefore be different. But this is impossible, since points A and A 1 lie on the same side of BC, and therefore BC can't pass through the midpoint of AA 1. Had it not been so, we would have had two isosceles triangles BAA 1 and CAA 1, and the perpendicular bisector of AA 1 would have to pass through points B and C in other words, the perpendicular bisector of AA 1 would have to coincide with the line BC. It would have been evident had line B'A' had the direction of BA or line C'A' the direction of CA. I claim that point A 1 coincides with point A. Let BCA 1 be the new position of triangle B'C'A'. Let's place the second triangle so as to make side B'C' coincide with BC and have both triangles lie on the same side of BC. Let there be two triangle ABC and A'B'C', whose sides are respectively equal. Now FG will either be in a straight line with DF, or make an angle with it, and in the latter case the angle will either be interior to the figure or exterior. Then C will coincide with F, since BC is equal to EF. Let the triangle ABC be applied to the triangle DEF, so that B is placed on E and BC on EF, but so that A falls on the opposite side EF from D, taking position G. Proclus gives the proof in the following order. Sir Thomas comments: This alternative proof avoids the use of I.7, and it is elegant but it is inconvenient in one respect, since three cases have to be distinguished. Sir Thomas Heath's commentary to Euclid contains a proof by the Hellenized Jewish philosopher Philo of Alexandria (or Philo Judeaus) that he plucked from Proclus' Commentary on the first book of Euclid. Therefore it is not possible that, if the base BC be applied to the base EF, the sides BA, AC should not coincide with ED, DF they will therefore coincide, so that the angle BAC will also coincide with the angle EDF, and be equal to it. Then, BC coinciding with EF, (so that) BA, AC wiil also coincide with ED, DF for, if the base BC coincides with the base EF, and the sides BA, AC do not coincide with ED, DF but fall beside them as EG, GF, then given two straight lines constructed on a straight line (from its extremities) and meeting in a point, there will have been constructed on the same straight line (from its extremities), and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each to that which has the same end with it. I say that the angle BAC is also equal to the angle EDF.įor, if the triangle ABC be applied to the triangle DEF, and if the point B be placed on the point E and the straight line BC on EF, the point C will also coincide with F, because BC is equal to EF.

Let ABC, DEF be two triangles having the two sides AB, AC equal to the two sides DE, DF respectively, namely AB to DE, and AC to DF and let them have the base BC equal to the base EF. On this page I have collected a few proofs of the SSS criterion from several popular geometry texts.

But he took care to make the two formulations sound alike so as to draw attention to the relationship between the two that he built and exploited. He did not find it necessary to mention that fact in a later proposition. Further, in Elements I.4, Euclid indeed concludes that under the SAS conditions, the remaining side and the remaining two angles in one triangle equal their counterparts in the other. As you'll see shortly, Euclid has based his proof of SSS on that of SAS. The reason for a strange, asymmetric formulation is that Elements I.4 is about what is known nowadays as SAS and establishes the congruence of two triangles by a pair of equal sides and the angle they include. Elements I.8 reads, If two triangles have the two sides equal to two sides respectively, and have also the base equal to the base, they will also have the angles equal which are contained by the equal straight lines. In Euclid's Elements the formulation is somewhat different. The acronym SSS (side-side-side) refers to the criterion of congruence of two triangles: if the three sides of one equal the three sides of the other, the two triangles are congruent.

Sas congruence