Is this question wrong? How can a circle be named ABC; it should be the triangle ABC, right?

Question

Let ABC be a triangle. Let S be the circle through B tangent to CA at A and let T be the circle through C tangent to AB at A. The circles S and T intersect at A and D. Let E be the point where the line AD meets the circle ABC. Prove that D is the midpoint of AE.

Answer 1

I hope this is correct...

Assuming a triangle $A,B,C$. it seems Circle $S$ has it's origin at the midpoint of the line $AB$ (radius $\cfrac{AB}{2}$). Similarly Circle $T$ has its origin at the midpoint of $BC$ (radius $\cfrac{BC}{2}$).

For the circles to intersect at $D$ means $AB=BC$ and $AE$ is the hypotenuse of the triangle $\Bigg(\bigg(\cfrac{AB}{2}\bigg)^{2}+\bigg(\cfrac{BC}{2}\bigg)^{2}\Bigg{)}^{1/2}$.

For $D$ to be the midpoint of $AE$. $D$ would need to be the origin of the circle $ABC$ with a radius of the hypotenuse above.