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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!

Note by Nihar Mahajan
1 year, 7 months ago

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The very next revolutionary book in the history of mathematics - Nihar's treatises on Euclidean Geometry. Swapnil Das · 1 year, 7 months ago

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@Swapnil Das 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 · 1 year, 7 months ago

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@Nihar Mahajan Do you have any other postulates which you have kept a secret? Swapnil Das · 1 year, 7 months ago

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@Swapnil Das I have one more ,I will be posting it soon on Brilliant. Actually , its the extension of this configuration. Nihar Mahajan · 1 year, 7 months ago

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@Nihar Mahajan It will be a success! Bye! Swapnil Das · 1 year, 7 months ago

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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 · 1 year, 7 months ago

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@Trevor Arashiro No the extension is something else. "Stay tuned" I will post it soon. Nihar Mahajan · 1 year, 7 months ago

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@Nihar Mahajan Mm, \(k^2\). You piqued my interest. Trevor Arashiro · 1 year, 7 months ago

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It's really nice seeing you discovering postulates at such an age. Swapnil Das · 1 year, 7 months ago

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@Swapnil Das 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 · 1 year, 7 months ago

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Congrats! xD Mehul Arora · 1 year, 7 months ago

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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 · 1 year, 7 months ago

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@Calvin Lin 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 · 1 year, 7 months ago

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@Nihar Mahajan Get rid of "right trapezoid such that ....", and list the important information as \( AB\perp BC \), \( BC \perp CD \). Calvin Lin Staff · 1 year, 7 months ago

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@Calvin Lin Thanks! I have edited it accordingly. Please tell more of your opinions about it (if any). Nihar Mahajan · 1 year, 7 months ago

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@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 · 1 year, 7 months ago

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@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 · 1 year, 7 months ago

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@Calvin Lin Oh , I see. Thanks for telling that.Lemme edit it. Nihar Mahajan · 1 year, 7 months ago

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@Calvin Lin @Azhaghu Roopesh M @Trevor Arashiro @Sharky Kesa Please see my discovery. Thanks! Nihar Mahajan · 1 year, 7 months ago

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