The Stars in the Water

But I'll try a different way, and maybe you'll see it better. Now mind, what I am to prove is this: if any triangle has two sides equal, the angles opposite those sides are also equal."

"And what difference does it make if they're equal or not?" said one of the men who stood near Kathleen.

"Be still there," the King said; "do we want to make telephones or do we not? And sure we can't make telephones without geometry. Hasn't Terence told you that?"

Terence went on: "Let ABC be any triangle in which the sides AB and AC are equal."

"How can it be any triangle, when it's only one triangle?" said another of the men.

"Keep your silly head shut," said the King. "Terence didn't say it was any triangle; he said let it be. Now will you let that triangle be, or will I come over there and make you let it be?"

The man said nothing more and Terence went on: "Now, consider this triangle as two triangles, BAC and CAB."

"How can it be two triangles," another of the men said, "when it's only one triangle?"

"Will you be still there?" the King said. "Terence doesn't say it's two triangles; he says you're to consider it. Will you consider that triangle two triangles, or will I come over there and make you consider it two triangles?"

"I'll consider it seven triangles, if you like, Your Majesty," the man answered, "but I dunno what good it'll do me."

"Then consider it," said the King, "and don't talk about it. Go on, Terence."

"Now, you see that since the sides AB and AC in each triangle are equal, AB and AC in the first are respectively equal to AC and AB in the second, and the angles between these sides are equal. So the two triangles are equal, by previous proposition. And so the angles of one are equal to the angles of the other, where they are opposite the equal sides; that is, the angle ABC is equal to the angle ACB, being opposite the equal sides AC and AB, by the same previous proposition, and that is what I was to prove."

The King looked at the men with triumph in his eye.