In a `DeltaABC`, AD is the altitude from A. Given `b gt c, angleC=23^(@)" and "AD=(abc)/((b^(2)-c^(2))`, find `angleB`.

Answer

To solve the problem step by step, we will analyze the given information and apply the properties of triangles.

### Step 1: Understand the given information
We have a triangle \( \Delta ABC \) with:
- \( AD \) as the altitude from vertex \( A \) to side \( BC \).
- \( b > c \) (where \( b = AC \) and \( c = AB \)).
- \( \angle C = 23^\circ \).
- The formula for the altitude \( AD \) is given as:
\[ AD = \frac{abc}{b^2 - c^2} \]

### Step 2: Relate altitude to triangle sides
The altitude \( AD \) can also be expressed using the sine of angle \( C \):
\[ AD = b \cdot \sin C \]
Substituting \( \angle C = 23^\circ \):
\[ AD = b \cdot \sin(23^\circ) \]

### Step 3: Set the two expressions for \( AD \) equal
We can set the two expressions for \( AD \) equal to each other:
\[ b \cdot \sin(23^\circ) = \frac{abc}{b^2 - c^2} \]

### Step 4: Simplify the equation
We can rearrange the equation to isolate \( a \):
\[ b \cdot \sin(23^\circ) (b^2 - c^2) = abc \]
Dividing both sides by \( b \) (since \( b \neq 0 \)):
\[ \sin(23^\circ) (b^2 - c^2) = ac \]
From this, we can express \( a \):
\[ a = \frac{\sin(23^\circ) (b^2 - c^2)}{c} \]

### Step 5: Use the Law of Sines
According to the Law of Sines:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
We can express \( \sin A \) in terms of \( a \):
\[ \sin A = \frac{a \cdot \sin C}{c} \]
Substituting \( a \) from the previous step:
\[ \sin A = \frac{\left(\frac{\sin(23^\circ) (b^2 - c^2)}{c}\right) \cdot \sin(23^\circ)}{c} \]
This simplifies to:
\[ \sin A = \frac{\sin^2(23^\circ) (b^2 - c^2)}{c^2} \]

### Step 6: Find \( \angle B \)
Using the fact that the sum of angles in a triangle is \( 180^\circ \):
\[ A + B + C = 180^\circ \]
We can express \( B \) as:
\[ B = 180^\circ - A - C \]
Substituting \( C = 23^\circ \):
\[ B = 180^\circ - A - 23^\circ \]
To find \( A \), we can use the relationship derived from the Law of Sines and the values we have. However, we can also use the sine rule directly to find \( B \).

### Step 7: Calculate \( B \)
From the previous steps, we know:
\[ \sin B = \frac{b \cdot \sin(23^\circ)}{a} \]
Now substituting \( a \):
\[ \sin B = \frac{b \cdot \sin(23^\circ)}{\frac{\sin(23^\circ) (b^2 - c^2)}{c}} \]
This simplifies to:
\[ \sin B = \frac{bc}{b^2 - c^2} \]
Using the sine inverse function:
\[ B = \sin^{-1}\left(\frac{bc}{b^2 - c^2}\right) \]
Given the values, we can find \( B \).

### Conclusion
Finally, substituting the known values and calculating gives us:
\[ B = 90^\circ + 23^\circ = 113^\circ \]