Prove Triangle BAD = CEA: Tips & Maths Class Help

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In summary: Yes, there is a better proof. You can see that the line $AB$ is tangent to the circle through $A$, $C$ and $D$ by using the Pythagorean theorem. Additionally, the triangles $BAD$, $CEA$ are congruent.
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Mathick
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In the triangle \(\displaystyle ABC\) a point \(\displaystyle D\) lies on the edge \(\displaystyle BC\), \(\displaystyle E\) - on the edge \(\displaystyle AB\). Aditionally, \(\displaystyle BD=AC\), \(\displaystyle AD=AE\) and \(\displaystyle AB^2=AC\cdot BC\). Prove that \(\displaystyle \sphericalangle BAD = \sphericalangle CEA\).

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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  • #2
Mathick said:
In the triangle \(\displaystyle ABC\) a point \(\displaystyle D\) lies on the edge \(\displaystyle BC\), \(\displaystyle E\) - on the edge \(\displaystyle AB\). Aditionally, \(\displaystyle BD=AC\), \(\displaystyle AD=AE\) and \(\displaystyle AB^2=AC\cdot BC\). Prove that \(\displaystyle \sphericalangle BAD = \sphericalangle CEA\).

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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  • #3
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 \(\displaystyle BD=AC\), \(\displaystyle AD=AE\) and \(\displaystyle AB^2=AC\cdot BC\), we get \(\displaystyle \frac{AB}{AC}=\frac{BC}{AB}\) and then \(\displaystyle \frac{AB}{BD}=\frac{BC}{AB}\). Thus, the triangles \(\displaystyle ABD\) and \(\displaystyle CBA\) are similar. So \(\displaystyle \sphericalangle BDA=\sphericalangle CAB\) and \(\displaystyle \sphericalangle BAD=\sphericalangle ACB\). Additionally, \(\displaystyle \frac{AC}{AB}=\frac{AD}{BD}\) (because the triangles \(\displaystyle ABD\) and \(\displaystyle CBA\) are similar). So \(\displaystyle \sphericalangle BAC=\sphericalangle BDA\), and what's more \(\displaystyle \sphericalangle ADC=\sphericalangle CA...\) (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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Related to Prove Triangle BAD = CEA: Tips & Maths Class Help

1. How do I prove that Triangle BAD is equal to Triangle CEA?

To prove that Triangle BAD is equal to Triangle CEA, you can use the Side-Angle-Side (SAS) postulate. This means that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent and therefore equal.

2. What are some tips for solving this proof?

One tip for solving this proof is to carefully label each angle and side in both triangles. This will help you keep track of which parts are equal and which ones you still need to prove. It's also important to use the given information in the problem, such as congruent angles or sides, to your advantage.

3. Can I use any other postulates or theorems to prove this statement?

Yes, besides the SAS postulate, you can also use the Side-Side-Side (SSS) postulate or the Angle-Side-Angle (ASA) postulate to prove that Triangle BAD is equal to Triangle CEA. Additionally, you can use the Congruent Triangles Theorem (CPCTC) to show that all corresponding parts of the triangles are equal.

4. What are the key steps in proving Triangle BAD = CEA?

The key steps in proving Triangle BAD = CEA include identifying the given information, labeling the triangles and their corresponding parts, using the appropriate postulate or theorem, and showing that all corresponding parts are equal. It's also important to write out each step clearly and logically to demonstrate your reasoning.

5. Can I use algebra to prove this statement?

Yes, you can use algebra to prove that Triangle BAD is equal to Triangle CEA. This can be done by setting up equations using the given information and using algebraic properties and rules to manipulate the equations until they are equal to each other. However, you still need to use a postulate or theorem to show that the triangles are congruent and equal.

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