The first error identified in the proof occurs in the step:
[tex]\[ z^2 = h^2 + r^2 \][/tex]
Here’s a detailed, step-by-step explanation leading up to this error and why it is incorrect:
1. Given:
Let \(\Delta XYZ\) be a triangle with an altitude \( h \) from vertex \( X \).
2. Define sine and cosine:
[tex]\[
\sin(X) = \frac{h}{z}, \quad \cos(X) = \frac{r}{z}
\][/tex]
By definitions of sine and cosine in a right triangle.
3. Multiplication property of equality:
[tex]\[
z \cdot \sin(X) = h, \quad z \cdot \cos(X) = r
\][/tex]
This follows directly from the definitions.
4. Substitution property of equality:
[tex]\[
x^2 = h^2 + (y - r)^2
\][/tex]
5. Substitution: Replace \( h \) and \( r \) using previous definitions:
[tex]\[
x^2 = (z \cdot \sin(X))^2 + (y - z \cdot \cos(X))^2
\][/tex]
6. Expand:
[tex]\[
x^2 = z^2 \sin^2(X) + y^2 - 2 y z \cos(X) + z^2 \cos^2(X)
\][/tex]
7. Factoring out:
[tex]\[
x^2 = z^2 [\sin^2(X) + \cos^2(X)] + y^2 - 2 y z \cos(X)
\][/tex]
8. Square both sides:
[tex]\[
\sin^2(X) = \frac{h^2}{z^2}, \quad \cos^2(X) = \frac{r^2}{z^2}
\][/tex]
9. Substitution:
[tex]\[
x^2 = z^2 \left( \frac{h^2}{z^2} + \frac{r^2}{z^2} \right) + y^2 - 2 y z \cos(X)
\][/tex]
10. Combine fractions:
[tex]\[
x^2 = z^2 \left( \frac{h^2 + r^2}{z^2} \right) + y^2 - 2 y z \cos(X)
\][/tex]
11. Error – Application of Pythagorean theorem:
[tex]\[
z^2 = h^2 + r^2
\][/tex]
The Pythagorean theorem \((a^2 + b^2 = c^2)\) is incorrectly applied here. It pertains to the sides of a right triangle, but in this context \( z^2 = h^2 + r^2 \) is not accurate as \( h \) and \( r \) typically correspond to different components of the triangle, not directly related by the Pythagorean theorem outside a specific right triangle setup.
Therefore, the step [tex]\( z^2 = h^2 + r^2 \)[/tex] is the first error in the proof, as it incorrectly applies the Pythagorean theorem to parameters that do not conform to its specific conditions.
