Difference between revisions of "User talk:Etmetalakret"

Revision as of 20:47, 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 AoPS one time. We let $\overline{BP}$ be an altitude of $\triangle ABC$ and hunt for triangle similarity. See the following diagram:

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 $c = AP + PB = \frac{a^2}{c} + \frac{b^2}{c}.$ Multiplying this equation by $c$ yields the desired $a^2 + b^2 = c^2$ .

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.

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: $c = AP + PB = \frac{a^2}{c} + \frac{b^2}{c}.$ 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:

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 $\frac{x + y}{2} \geq \sqrt{xy}.$ (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.)

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. Really understand and grasp how I derived this before you read the following proofs:

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

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. $\square$

@@ Line 1: / Line 1: @@
 AoPS Wiki users, ignore this page. I'm using my User Talk to explain proof writing to friends.
-== Proof 1: Inequalities ==
+== Proof 1: The Pythagorean Theorem ==
+Prove the Pythagorean Theorem for a right triangle <math>\triangle ABC</math> such that <math>\angle BCA = 90^{\circ}</math>.
+=== 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:
+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>.
+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''.
+=== 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>
+'''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 <math>a</math>, <math>b</math>, and <math>c</math>? What about <math>\angle B</math>? You can still use these these instead of writing out <math>BC</math>, <math>CA</math>, <math>AB</math>, and <math>\angle ABC</math>, but you need to define them.
+* '''Not Enough Space''': GIVE THESE WALLS OF EQUATIONS THEIR OWN LINES!
+* '''Degrees Not Specified''': The proof references <math>90^{\circ}</math>, not whatever <math>90</math> means.
+=== 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:
+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 ==
 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.)
@@ Line 22: / Line 46: @@
 * '''Written Backwards''': We must always write proofs like: true result <math>\implies</math> desired result. However, the proof is written backwards so that the desired result <math>\implies</math> 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 <math>2</math> 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 be given 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 ===
-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 the desired inequality of <math>(x + y) / 2 \geq \sqrt{xy}</math>. <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>

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Difference between revisions of "User talk:Etmetalakret"

Revision as of 20:47, 1 April 2023

Contents

Proof 1: The Pythagorean Theorem

Explanation

Bad Proof

Good Proof

Proof 2: Inequalities

Explanation

Bad Proof

Good Proof