Formal Exploration of Geometry

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A formal exploration of constructive geometry

Proposition 4

If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend.

This proposition is essentially the side-angle-side property, which states that if two sides and the angle between them are congruent to two sides and the angle between them of another triangle, then the triangles are congruent.

Euclid's proof of this proposition is not constructive and hardly even logical; he solves this proof by applying one triangle to the other. This modus operandi has many fallacies, such that our triangles are not material and we can not simply pick one up and place it on the other for comparison. Even if they were able to be picked up and moved, we would have to worry about the rigidity of the triangle being moved so that the shape is not deformed in any way.

Euclid abuses properties of superposition to 'apply' one triangle to the other to verify their congruency without further definition or explanation of this triangle application method. This is simply not a valid method of geometric proof and holds no mathematical content; if this method were valid we would solve most all of the rest in a similar fashion. Euclid proves Propositions 6 and 7 similarly, which is why the constructive proof is not obvious.

The only support he uses in his proof is Common Notion 4 that says, "Things which coincide with one another are equal to one another." This common notion is just as vacuous as applying a triangle because Euclid doesn't elaborate on what it means for two things to 'coincide'.

The diagram below may seem trivial because when you move points A, B, or C, points D, E, and F will change respectively. There are no steps to this diagram because it is given that 2 sides and the angle they create of ABC are equivalent to 2 sides and angles of DEF, which we know means the triangles are congruent. If we change any length of any side of ABC, the change happens in DEF to maintain the assumptions of equality, which always maintains congruent triangles.

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 the angle BAC equal to the angle EDF.

I say that the base BC is also equal to the base EF, the triangle ABC will be equal to the triangle DEF, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend, that is, the angle ABC to the angle DEF, and the angle ACB to the angle DFE.

For, if the triangle ABC be applied to the triangle DEF, and if the point A be placed on the point D and the straight line AB on DE, then the point B will also coincide with E, because AB is equal to DE.

Again, AB coinciding with DE, the straight line AC will also coincide with DF, because the angle BAC is equal to the angle EDF; hence the point C will also coincide with the point F, because AC is again equal to EF.

But B also coincided with E; hence the base BC will coincide with the base EF, and will be equal to it. [C.N.4]Things which coincide with one another are equal to one another.

Thus the whole triangle ABC will coincide whole triangle DEF, and will be equal to it. [C.N.4]Things which coincide with one another are equal to one another. And the remaining angles will be equal to them, the angle ABC to the angle DEF, and the angle ACB to the angle DFE. [C.N.4]Things which coincide with one another are equal to one another.

Therefore etc.

Q.E.D.

Proof in Nuprl

Index