I discovered a TONCAS 1

I made a simple configuration , whose converse is also true. Though the proof is very easy , the important thing is that its converse is also true!

Problem statement: Let $AB \perp BC$ , $DC \perp BC$ . Consider a point $P$ inside $ABCD$ . Let $P_1$ be its reflection in $\overline{AB}$ and $P_2$ be its reflection in $\overline{CD}$ . Let $\overrightarrow{P_1B} \cap \overrightarrow{P_2C} = {P_3}$ . Prove that $\overline{P_3P} \perp \overline{P_1P_2}$ .

Statement converse: Consider a quadrilateral $ABCD$ . Consider a point $P$ inside $ABCD$ . Let $P_1$ be its reflection in $\overline{AB}$ and $P_2$ be its reflection in $\overline{CD}$ . Let $\overrightarrow{P_1B} \cap \overrightarrow{P_2C} ={P_3}$ . If $\overline{P_3P} \perp \overline{P_1P_2}$ , then prove that $AB \perp BC$ , $DC \perp BC$ .

I have my own solution too. Please post awesome "complete" solutions below. Enjoy!

#Geometry

Easy Math Editor

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Comments

The very next revolutionary book in the history of mathematics - Nihar's treatises on Euclidean Geometry.

Swapnil Das - 6 years ago

Well , I think you over-understood me as the best in geometry. I am just "good" at geometry since its one of my interests. Anyway , thanks!

Nihar Mahajan - 6 years ago

Do you have any other postulates which you have kept a secret?

Swapnil Das - 6 years ago

@Swapnil Das – I have one more ,I will be posting it soon on Brilliant. Actually , its the extension of this configuration.

Nihar Mahajan - 6 years ago

@Nihar Mahajan – It will be a success! Bye!

Swapnil Das - 6 years ago

@Calvin Lin @Azhaghu Roopesh M @Trevor Arashiro @Sharky Kesa Please see my discovery. Thanks!

Nihar Mahajan - 6 years ago

Since points A and D are "essentially useless" other than saying that we have parallel / perpendicular lines, you should remove them from the statement.

Calvin Lin Staff - 6 years ago

Uum ... I have points $A,D$ just for labeling the angles , segments , that is for notation convenience. I am not able to understand why are you saying to remove them.

Nihar Mahajan - 6 years ago

Get rid of "right trapezoid such that ....", and list the important information as $AB\perp BC$ , $BC \perp CD$ .

Calvin Lin Staff - 6 years ago

@Calvin Lin – Oh , I see. Thanks for telling that.Lemme edit it.

Nihar Mahajan - 6 years ago

@Calvin Lin – Thanks! I have edited it accordingly. Please tell more of your opinions about it (if any).

Nihar Mahajan - 6 years ago

@Nihar Mahajan – There is no point in saying "Join AD" is there? The idea is to remove irrelevant information like that, so that you're left with just the important attributes. This will make it easier to apply in other scenarios (where there isn't clearly a rectangular trapezoid).

Calvin Lin Staff - 6 years ago

@Calvin Lin – Oh , I completely understood what your intention was. I would definitely take care about this when I will post the extension of this configuration soon. "stay tuned" :P

Nihar Mahajan - 6 years ago

Congrats! xD

Mehul Arora - 6 years ago

It's really nice seeing you discovering postulates at such an age.

Swapnil Das - 6 years ago

Actually this is not a postulate. This is a simple configuration whose converse is also true. Postulates/axioms are just defined and not proved. They are "used" to prove things.

Nihar Mahajan - 6 years ago

What's a TONCAS 1?

Btw, the proof is simple for the positive statement. Just uses scale factors and similar triangles. And by proving the positive, the converse is proved here. I'm curious as to what your extension is. At any guess, I'd say reflecting a point about two parallel lines and having the three points form a right triangle about the intersection of the two lines.

Trevor Arashiro - 6 years ago

No the extension is something else. "Stay tuned" I will post it soon.

Nihar Mahajan - 6 years ago

Mm, $k^2$ . You piqued my interest.

Trevor Arashiro - 6 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$