Bisect AC so that AE and CE are equal.

Let ABC be the given triangle.

Extend (produce) the side BC to BD.

We will show that angle DCA is greater than either angle CBA or CAB.

 

Construct BE and extend BE to BF so that BE and BF are equal.

Construct FC.

We will show that triangles AEB and CEF are congruent.

Show me.

 

(CN-x)

(Prop I-xx)

(Post-xx)

We already know that AE equals CE and that BE equals FE.

We also know that angle AEB equals angle CEF.
(Prop I-15)

Therefore, the congruence of AEB and CEF is proved.
(Prop I-4)

It follows that angle EAB equals angle ECF, and finally that angle ECF is less than angle ACD.

Similarly, bisecting BC, one proves that angle CBA is less than angle ACD.

QED