Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "User talk:Etmetalakret"
Line 5: Line 5:
  
 
=== Explanation ===
 
=== Explanation ===
−
I showed this proof in AoPS one time. We let <math>\overline{BP}</math> be an altitude of <math>\triangle ABC</math> and hunt for triangle similarity. See the following diagram:
+
I showed this proof in the study group one time. We let <math>\overline{BP}</math> be an altitude of <math>\triangle ABC</math> and hunt for triangle similarity. See the following diagram:
  
−
Examine the triangles <math>\triangle CBP</math> and <math>\triangle ABC</math>. They both share <math>\angle B</math> and a right angle, so AA Similarity guarantees that <math>\triangle CBP \sim \triangle ABC</math>. Similarly, <math>\triangle ACP \sim \triangle ABC</math>. We thus get the following ratios: <cmath>\frac{PB}{a} = \frac{a}{c} \textrm{ and } \frac{AP}{b} = \frac{b}{c}.</cmath> We can solve for <math>PB</math> and <math>AP</math> as follows: <cmath>PB = \frac{a^2}{c} \textrm{ and } AP = \frac{b^2}{c}.</cmath> But why is this useful? It's because <math>AP + PB = c</math>. Using this fact, we have that <cmath>c = AP + PB = \frac{a^2}{c} + \frac{b^2}{c}.</cmath> Multiplying this equation by <math>c</math> yields the desired <math>a^2 + b^2 = c^2</math>.
+
[[Image:Screenshot 2023-04-01 204908.png|thumb|none|600px]]
  
−
It's important to acknowledge that the above is not a well-written proof. Your goal in a proof is NOT to show how you found it; it's simply to ''illustrate why it's true''.
+
We let <math>BC = a</math>, <math>CA = b</math>, and <math>AB = c</math>.
 +
 
 +
Examine the triangles <math>\triangle CBP</math> and <math>\triangle ABC</math>. They both share <math>\angle B</math> and a right angle, so AA Similarity guarantees that <math>\triangle CBP \sim \triangle ABC</math>. Similarly, <math>\triangle ACP \sim \triangle ABC</math>. We thus get the following ratios: <cmath>\frac{PB}{a} = \frac{a}{c} \textrm{ and } \frac{AP}{b} = \frac{b}{c}.</cmath> We can solve for <math>PB</math> and <math>AP</math> as follows: <cmath>PB = \frac{a^2}{c} \textrm{ and } AP = \frac{b^2}{c}.</cmath> But why is this useful? It's because <math>AP + PB = c</math>. Using this fact, we have that <cmath>c = AP + PB = \frac{a^2}{c} + \frac{b^2}{c} = \frac{a^2 + b^2}{c}.</cmath> Multiplying this equation by <math>c</math> yields the desired <math>a^2 + b^2 = c^2</math>.
  
 
=== Bad Proof ===  
 
=== Bad Proof ===  
−
Note that <math>\angle PBC = \angle CBA = \angle B</math> and <math>\angle BPC = BCA = 90</math>. We thus have by AA Similarity that <math>\triangle CBP \sim \triangle ABC</math>. Similarly, <math>\triangle ACP \sim \triangle ABC</math>. Therefore, <math>\frac{PB}{a} = \frac{a}{c} \textrm{ and } \frac{AP}{b} = \frac{b}{c}.</math> We can solve for <math>PB</math> and <math>AP</math> as follows: <math>PB = \frac{a^2}{c} \textrm{ and } AP = \frac{b^2}{c}.</math> Then the following sequence of equations holds: <math>c = AP + PB = \frac{a^2}{c} + \frac{b^2}{c}.</math> Multiplying this equation by <math>c</math> yields the desired <math>a^2 + b^2 = c^2</math>. <math>\square</math>
+
Note that <math>\angle PBC = \angle CBA = \angle B</math> and <math>\angle BPC = BCA = 90</math>. We thus have by AA Similarity that <math>\triangle CBP \sim \triangle ABC</math>. Similarly, <math>\triangle ACP \sim \triangle ABC</math>. Therefore, <math>\frac{PB}{a} = \frac{a}{c} \textrm{ and } \frac{AP}{b} = \frac{b}{c}.</math> We can solve for <math>PB</math> and <math>AP</math> as follows: <math>PB = \frac{a^2}{c} \textrm{ and } AP = \frac{b^2}{c}.</math> Then the following sequence of equations holds: <math>c = AP + PB = \frac{a^2}{c} + \frac{b^2}{c} + \frac{a^2 + b^2}{c}.</math> Multiplying this equation by <math>c</math> yields the desired <math>a^2 + b^2 = c^2</math>. <math>\square</math>
  
 
'''Why is this proof bad?'''
 
'''Why is this proof bad?'''
Line 22: Line 24:
 
=== Good Proof ===
 
=== Good Proof ===
 
Let <math>P</math> be the point on <math>\overline{AB}</math> such that <math>\angle BPC = 90^{\circ}</math>, as shown in the following diagram:
 
Let <math>P</math> be the point on <math>\overline{AB}</math> such that <math>\angle BPC = 90^{\circ}</math>, as shown in the following diagram:
 +
 +
[[Image:Screenshot 2023-04-01 204908.png|thumb|none|600px]]
  
 
Note that <math>\angle PBC = \angle CBA</math> and <math>\angle BPC = BCA = 90^{\circ}</math>. We thus have by AA Similarity that <math>\triangle CBP \sim \triangle ABC</math>. Similarly, <math>\triangle ACP \sim \triangle ABC</math>. Therefore, <cmath>\frac{PB}{BC} = \frac{CB}{BA} \textrm{ and } \frac{AP}{AC} = \frac{AC}{AB}.</cmath> We can solve for <math>PB</math> and <math>AP</math> as follows: <cmath>PB = \frac{BC^2}{AB} \textrm{ and } AP = \frac{CA^2}{AB}.</cmath> Then the following sequence of equations holds: <cmath>AB = AP + PB = \frac{BC^2}{AB} + \frac{CA^2}{AB} = \frac{BC^2 + CA^2}{AB}.</cmath> Multiplying this equation by <math>AB</math> yields the desired <math>BC^2 + CA^2 = AB^2</math>. <math>\square</math>
 
Note that <math>\angle PBC = \angle CBA</math> and <math>\angle BPC = BCA = 90^{\circ}</math>. We thus have by AA Similarity that <math>\triangle CBP \sim \triangle ABC</math>. Similarly, <math>\triangle ACP \sim \triangle ABC</math>. Therefore, <cmath>\frac{PB}{BC} = \frac{CB}{BA} \textrm{ and } \frac{AP}{AC} = \frac{AC}{AB}.</cmath> We can solve for <math>PB</math> and <math>AP</math> as follows: <cmath>PB = \frac{BC^2}{AB} \textrm{ and } AP = \frac{CA^2}{AB}.</cmath> Then the following sequence of equations holds: <cmath>AB = AP + PB = \frac{BC^2}{AB} + \frac{CA^2}{AB} = \frac{BC^2 + CA^2}{AB}.</cmath> Multiplying this equation by <math>AB</math> yields the desired <math>BC^2 + CA^2 = AB^2</math>. <math>\square</math>
  
 
== Proof 2: Inequalities ==
 
== Proof 2: Inequalities ==
−
The well-known '''Trivial Inequality''' states that if <math>x</math> is a real number, then <math>x^2 \geq 0</math>. Prove that if <math>x</math> and <math>y</math> are nonnegative real numbers, then <cmath>\frac{x + y}{2} \geq \sqrt{xy}.</cmath> (Sidenote: this is a very different kind of inequality problem than you're used to. In school, we find ''when'' inequalities are true; here, we're showing it's ''always'' true.)
+
The well-known '''Trivial Inequality''' states that if <math>x</math> is a real number, then <math>x^2 \geq 0</math>. Prove that if <math>x</math> and <math>y</math> are nonnegative real numbers, then <cmath>\frac{x + y}{2} \geq \sqrt{xy}.</cmath>  
  
 
=== Explanation ===
 
=== Explanation ===
Line 38: Line 42:
 
(x - y)^2 \geq 0.
 
(x - y)^2 \geq 0.
 
\end{align*}</cmath>
 
\end{align*}</cmath>
−
Because the left-hand side of this equation is a perfect square, this is actually the Trivial Inequality in disguise. The desired inequality is therefore implied by a true result. Really understand and grasp how I derived this before you read the following proofs:
+
Because the left-hand side of this equation is a perfect square, this is actually the Trivial Inequality in disguise. The desired inequality is therefore implied by a true result. We can now write a proof:
  
 
=== Bad Proof ===
 
=== Bad Proof ===
Line 48: Line 52:
 
* '''Not Enough Space''': A little more space would make this proof easier to read. Important equations should have their own line.
 
* '''Not Enough Space''': A little more space would make this proof easier to read. Important equations should have their own line.
  
−
=== Good Proof ===
+
=== Good Proof 1 ===
 
By the Trivial Inequality, we have that <cmath>(x - y)^2 \geq 0.</cmath> Factoring this inequality returns <math>x^2 - 2xy + y^2 \geq 0</math>. We add <math>4xy</math> to both sides and factor to get <math>(x + y)^2 \geq 4xy</math>. Note that because <math>x</math> and <math>y</math> are nonnegative, both sides are nonnegative; we may therefore take the square root of the inequality, which yields <cmath>x + y \geq 2 \sqrt{xy}.</cmath> Finally, dividing both sides by <math>2</math> gives <math>(x + y) / 2 \geq \sqrt{xy}</math>, which completes the proof. <math>\square</math>
 
By the Trivial Inequality, we have that <cmath>(x - y)^2 \geq 0.</cmath> Factoring this inequality returns <math>x^2 - 2xy + y^2 \geq 0</math>. We add <math>4xy</math> to both sides and factor to get <math>(x + y)^2 \geq 4xy</math>. Note that because <math>x</math> and <math>y</math> are nonnegative, both sides are nonnegative; we may therefore take the square root of the inequality, which yields <cmath>x + y \geq 2 \sqrt{xy}.</cmath> Finally, dividing both sides by <math>2</math> gives <math>(x + y) / 2 \geq \sqrt{xy}</math>, which completes the proof. <math>\square</math>
 +
 +
=== Good Proof 2 ===
 +
By the Trivial Inequality, we have that <cmath>(x - y)^2 \geq 0.</cmath> Then the following sequence of inequalities holds:
 +
<cmath>
 +
\begin{align*}
 +
x^2 - 2xy + y^2 \geq 0 \\
 +
x^2 + 2xy + y^2 \geq 4xy \\
 +
(x + y)^2 \geq 4xy.
 +
\end{align*}</cmath>
 +
Note that because <math>x</math> and <math>y</math> are nonnegative, both sides of this final inequality are nonnegative; we may therefore take the square root of both sides, which yields <cmath>x + y \geq 2 \sqrt{xy}.</cmath> Finally, dividing the inequality by <math>2</math> gives <math>(x + y) / 2 \geq \sqrt{xy}</math>, which completes the proof. <math>\square</math>

Revision as of 21:01, 1 April 2023

AoPS Wiki users, ignore this page. I'm using my User Talk to explain proof writing to friends.

Proof 1: The Pythagorean Theorem

Prove the Pythagorean Theorem for a right triangle $\triangle ABC$ such that $\angle BCA = 90^{\circ}$.

Explanation

I showed this proof in the study group one time. We let $\overline{BP}$ be an altitude of $\triangle ABC$ and hunt for triangle similarity. See the following diagram:

Screenshot 2023-04-01 204908.png

We let $BC = a$, $CA = b$, and $AB = c$.

Examine the triangles $\triangle CBP$ and $\triangle ABC$. They both share $\angle B$ and a right angle, so AA Similarity guarantees that $\triangle CBP \sim \triangle ABC$. Similarly, $\triangle ACP \sim \triangle ABC$. We thus get the following ratios: \[\frac{PB}{a} = \frac{a}{c} \textrm{ and } \frac{AP}{b} = \frac{b}{c}.\] We can solve for $PB$ and $AP$ as follows: \[PB = \frac{a^2}{c} \textrm{ and } AP = \frac{b^2}{c}.\] But why is this useful? It's because $AP + PB = c$. Using this fact, we have that \[c = AP + PB = \frac{a^2}{c} + \frac{b^2}{c} = \frac{a^2 + b^2}{c}.\] Multiplying this equation by $c$ yields the desired $a^2 + b^2 = c^2$.

Bad Proof

Note that $\angle PBC = \angle CBA = \angle B$ and $\angle BPC = BCA = 90$. We thus have by AA Similarity that $\triangle CBP \sim \triangle ABC$. Similarly, $\triangle ACP \sim \triangle ABC$. Therefore, $\frac{PB}{a} = \frac{a}{c} \textrm{ and } \frac{AP}{b} = \frac{b}{c}.$ We can solve for $PB$ and $AP$ as follows: $PB = \frac{a^2}{c} \textrm{ and } AP = \frac{b^2}{c}.$ Then the following sequence of equations holds: $c = AP + PB = \frac{a^2}{c} + \frac{b^2}{c} + \frac{a^2 + b^2}{c}.$ Multiplying this equation by $c$ yields the desired $a^2 + b^2 = c^2$. $\square$

Why is this proof bad?

  • No Diagram: You ALWAYS need a diagram in geometry proofs to help the grader remain oriented in dense notation.
  • Terms have not been defined: What are $a$, $b$, and $c$? What about $\angle B$? You can still use these these instead of writing out $BC$, $CA$, $AB$, and $\angle ABC$, but you need to define them.
  • Not Enough Space: GIVE THESE WALLS OF EQUATIONS THEIR OWN LINES!
  • Degrees Not Specified: The proof references $90^{\circ}$, not whatever $90$ means.

Good Proof

Let $P$ be the point on $\overline{AB}$ such that $\angle BPC = 90^{\circ}$, as shown in the following diagram:

Screenshot 2023-04-01 204908.png

Note that $\angle PBC = \angle CBA$ and $\angle BPC = BCA = 90^{\circ}$. We thus have by AA Similarity that $\triangle CBP \sim \triangle ABC$. Similarly, $\triangle ACP \sim \triangle ABC$. Therefore, \[\frac{PB}{BC} = \frac{CB}{BA} \textrm{ and } \frac{AP}{AC} = \frac{AC}{AB}.\] We can solve for $PB$ and $AP$ as follows: \[PB = \frac{BC^2}{AB} \textrm{ and } AP = \frac{CA^2}{AB}.\] Then the following sequence of equations holds: \[AB = AP + PB = \frac{BC^2}{AB} + \frac{CA^2}{AB} = \frac{BC^2 + CA^2}{AB}.\] Multiplying this equation by $AB$ yields the desired $BC^2 + CA^2 = AB^2$. $\square$

Proof 2: Inequalities

The well-known Trivial Inequality states that if $x$ is a real number, then $x^2 \geq 0$. Prove that if $x$ and $y$ are nonnegative real numbers, then \[\frac{x + y}{2} \geq \sqrt{xy}.\]

Explanation

I found the proof by working backwards; I started with the desired result, and connected it to something true. Here is the wall of equations on my page (sadly I can't get them aligned): \begin{align*} \frac{x + y}{2} \geq \sqrt{xy} \\ x + y \geq 2 \sqrt{xy} \\ (x + y)^2 \geq 4xy \\ x^2 + 2xy + y^2 \geq 4xy \\ x^2 - 2xy + y^2 \geq 0 \\ (x - y)^2 \geq 0. \end{align*} Because the left-hand side of this equation is a perfect square, this is actually the Trivial Inequality in disguise. The desired inequality is therefore implied by a true result. We can now write a proof:

Bad Proof

I start out with $\frac{x + y}{2} \geq \sqrt{xy}.$ Multiply the inequality by $2$ and square it, $(x + y)^2 \geq 2 \sqrt{xy}$. Letting our algebra go on autopilot, $x^2 + 2xy + y^2 \geq 4xy$ and $x^2 - 2xy + y^2 \geq 0$, so $(x - y)^2 \geq 0$. This is true by Trivial Inequality, which completes the proof. $\square$

Why is this proof bad?

  • Written Backwards: We must always write proofs like: true result $\implies$ desired result. However, the proof is written backwards so that the desired result $\implies$ true result. The Trivial Inequality should be at the start, not the end.
  • Informal Word Choice: Please don't use the phrase "algebra autopilot" in a proof, and don't write sentences with no verbs (see the "Multiply the inequality by $2$ and square it"). Also, don't use "I," although "we" is totally acceptable.
  • Not Enough Space: A little more space would make this proof easier to read. Important equations should have their own line.

Good Proof 1

By the Trivial Inequality, we have that \[(x - y)^2 \geq 0.\] Factoring this inequality returns $x^2 - 2xy + y^2 \geq 0$. We add $4xy$ to both sides and factor to get $(x + y)^2 \geq 4xy$. Note that because $x$ and $y$ are nonnegative, both sides are nonnegative; we may therefore take the square root of the inequality, which yields \[x + y \geq 2 \sqrt{xy}.\] Finally, dividing both sides by $2$ gives $(x + y) / 2 \geq \sqrt{xy}$, which completes the proof. $\square$

Good Proof 2

By the Trivial Inequality, we have that \[(x - y)^2 \geq 0.\] Then the following sequence of inequalities holds: \begin{align*} x^2 - 2xy + y^2 \geq 0 \\ x^2 + 2xy + y^2 \geq 4xy \\ (x + y)^2 \geq 4xy. \end{align*} Note that because $x$ and $y$ are nonnegative, both sides of this final inequality are nonnegative; we may therefore take the square root of both sides, which yields \[x + y \geq 2 \sqrt{xy}.\] Finally, dividing the inequality by $2$ gives $(x + y) / 2 \geq \sqrt{xy}$, which completes the proof. $\square$

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=User_talk:Etmetalakret&oldid=191694"