MHB Prove Triangle BAD = CEA: Tips & Maths Class Help

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To prove that angles BAD and CEA are equal in triangle ABC, it is essential to establish the congruence of triangles BAD and CEA using the given conditions: BD equals AC, AD equals AE, and AB squared equals AC times BC. The proof involves demonstrating that line AB is tangent to the circumcircle of triangle ACD, which can be shown through the similarity of triangles ABD and CBA. This similarity leads to the conclusion that angles BDA and CAB are equal, ultimately resulting in the equality of angles BAD and CEA. The discussion seeks further validation of the proof's correctness and potential alternative methods.
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In the triangle $$ABC$$ a point $$D$$ lies on the edge $$BC$$, $$E$$ - on the edge $$AB$$. Aditionally, $$BD=AC$$, $$AD=AE$$ and $$AB^2=AC\cdot BC$$. Prove that $$\sphericalangle BAD = \sphericalangle CEA$$.

I have to do this task on Maths class on Monday and I need any tips or something because I don't know how to start it.
 
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Mathick said:
In the triangle $$ABC$$ a point $$D$$ lies on the edge $$BC$$, $$E$$ - on the edge $$AB$$. Aditionally, $$BD=AC$$, $$AD=AE$$ and $$AB^2=AC\cdot BC$$. Prove that $$\sphericalangle BAD = \sphericalangle CEA$$.

I have to do this task on Maths class on Monday and I need any tips or something because I don't know how to start it.
I won't give you the answer, but see if you can show that (1) the line $AB$ is tangent to the circle through $A$, $C$ and $D$; (2) the triangles $BAD$, $CEA$ are congruent.
 

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Opalg said:
I won't give you the answer, but see if you can show that (1) the line $AB$ is tangent to the circle through $A$, $C$ and $D$; (2) the triangles $BAD$, $CEA$ are congruent.

Proof: the line $AB$ is tangent to the circle through $A$, $C$ and $D$

Knowing that $$BD=AC$$, $$AD=AE$$ and $$AB^2=AC\cdot BC$$, we get $$\frac{AB}{AC}=\frac{BC}{AB}$$ and then $$\frac{AB}{BD}=\frac{BC}{AB}$$. Thus, the triangles $$ABD$$ and $$CBA$$ are similar. So $$\sphericalangle BDA=\sphericalangle CAB$$ and $$\sphericalangle BAD=\sphericalangle ACB$$. Additionally, $$\frac{AC}{AB}=\frac{AD}{BD}$$ (because the triangles $$ABD$$ and $$CBA$$ are similar). So $$\sphericalangle BAC=\sphericalangle BDA$$, and what's more $$\sphericalangle ADC=\sphericalangle CA...$$ (there is no letter on line AB after A).

As a result, the line AB is tangent to the circle through $A$, $C$ and $D$ because of the theorem of which is illustrated on the photo.

View attachment 4735

Is it correct? Is there any better proof?
 

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