Proof of the isosceles triangle theorem

The Isosceles Triangle Theorem
Suppose that ABC is an h-triangle.
(1) If d(A,B) = d(A,C) then <ABC = <ACB.
(2) If <ABC = <ACB then d(A,B) = d(A,C).

Consider the h-triangles ABC and ACB.
The order of vertices is significant, so we use different colours!

(1) We have
d(A,B) = d(A,C),
d(A,C) = d(A,B),
d(A,B) = d(A,C),
Thus, the h-triangles are h-congruent by the (SSS) condition.
It follows that <ABC = <ACB.

Is similar, using the (ASA) condition; we leave it as an exercise.

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