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 Subject: "New Pythagorean Theorem Proof ?" Previous Topic | Next Topic
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Bui Quang Tuan
Member since Jun-23-07
Aug-17-08, 01:53 PM (EST)    "New Pythagorean Theorem Proof ?"

 Dear All My Friends,Suppose ABC is right triangle with side lengths a, b, c and AB = c is hypotenuse. CH is altitude from C to AB. Denote (XYZ) = area of polygon XYZ.We translate ABC by direction HC to get triangle A'B'C' (not necessary exact how is CC').A1 = intersection of AC, B'C'B1 = intersection of BC, A'C'D = intersection of BC, AA'D' = intersection of B'C', AA'By some parallelograms we can show:(ACB1) = (ACC') = (DCC') = (DCC'D')/2(BCA1) = (BCC') = (BCC'B')/2(ABB') = (ABB'A')/2 = (BDD'B')/2therefore:(ABB') = (ACB1) + (BCA1) (1)By some right angles we can show that A1, B1, A, A', B, B' are on circumcircle of rectangle ABB'A' therefore:angle(BAB') = angle(BA1B') = angle(CBA1) = angle(CAB1)It means three right triangles ABB', BCA1, CAB1 are similar. Suppose k = BB'/AB = CA1/BC = CB1/CA then (1) can be written:k*c^2 = k*b^2 + k*a^2 or we have Pythagorean theorem:c^2 = b^2 + a^2May be it can be new proof for Pythagorean theorem?Thank you and best regards,Bui Quang Tuan

Subject     Author     Message Date     ID New Pythagorean Theorem Proof ? Bui Quang Tuan Aug-17-08 TOP RE: New Pythagorean Theorem Proof ? alexb Aug-17-08 1 RE: New Pythagorean Theorem Proof ? Bui Quang Tuan Aug-17-08 2 RE: New Pythagorean Theorem Proof ? Bui Quang Tuan Aug-18-08 3 RE: New Pythagorean Theorem Proof ? alexb Aug-18-08 4 RE: New Pythagorean Theorem Proof ? Bui Quang Tuan Aug-18-08 5 RE: New Pythagorean Theorem Proof ? Bui Quang Tuan Aug-19-08 6

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alexb
Charter Member
2261 posts
Aug-17-08, 02:11 PM (EST)    1. "RE: New Pythagorean Theorem Proof ?"
In response to message #0

 I am quite positive the proof is not listed in Loomis' book. It is somewhat bulky to my liking. Also, it is strange to eventually use similarity of triangles after skipping the similarily of ABC, ACH and BCH.I attach another proof that I received recently. Bui Quang Tuan
Member since Jun-23-07
Aug-17-08, 07:52 AM (EST)    2. "RE: New Pythagorean Theorem Proof ?"
In response to message #1

 Dear Alex,I have found this proof when I try to prove one area property of orthodiagonal inscribed quadrilateral. By my notations, it is quadrilateral ABA1B1 inscribed in circle, say (O). The property is:area of (ABO) is arithmetic mean of areas of (ACB1) and (BCA1). I use same method to prove this fact.After the proof I see three similar right triangles on sides of right triangle ABC and by using this similarity I found the proof for Pythagorean triangle ABC.Instead of translation we can also start by construction any circle (O) passing B, C and cutting extent AC, BC at A1, B1.Best regards,Bui Quang Tuan Bui Quang Tuan
Member since Jun-23-07
Aug-18-08, 08:08 PM (EST)    3. "RE: New Pythagorean Theorem Proof ?"
In response to message #2

 Dear Alex,By using this area property of orthodiagonal inscribed quadrilateral we can also prove Pythagorean theorem. Detail as following:ABC is a right triangle with right angle at C and a, b, c are side lengths.Extent BC and choose CA' = CA.Extent AC and choose CB' = CB.ABB'A' is isosceles trapezium and it is orthodiagonal quadrilateral inscribed in a circle say (O). We have: ACA', BCB', OAB are right isosceles triangles. By area property of orthodiagonal inscribed quadrilateral:area(ACA') + area(BCB') = 2*area(OAB)b^2/2 + a^2/2 = 2*c^2/4b^2 + a^2 = c^2(Please see attach picture file).Best regards,Bui Quang Tuan alexb
Charter Member
2261 posts
Aug-18-08, 11:11 PM (EST)    4. "RE: New Pythagorean Theorem Proof ?"
In response to message #3

 Is it really different fromhttps://www.cut-the-knot.org/pythagoras/Orthodiagonal2.shtml? Bui Quang Tuan
Member since Jun-23-07
Aug-18-08, 07:17 AM (EST)    5. "RE: New Pythagorean Theorem Proof ?"
In response to message #4

 Dear Alex,At the first sight, I think it is really different because:- My property is for orthodiagonal and inscribed quadrilateral.- I use circumcenter of this quadrilateral.In any case, I will see it carefully today and reply with more detail.Best regards,Bui Quang Tuan Bui Quang Tuan
Member since Jun-23-07
Aug-19-08, 07:17 AM (EST)    6. "RE: New Pythagorean Theorem Proof ?"
In response to message #4

 Dear Alex,Of course my proof is different because:- My area property of orthodiagonal inscribed quadrilateral is different- My proof is based on general fact when k = 1- It is very clear if we draw proof picture when k = 1 (please see attach file)But when draw this picture I see it is exactly proof 69 "Pythagoras' Theorem By Sheer Shearing":https://www.cut-the-knot.org/pythagoras/PyhtI47PWW.shtmlWhen look again generalization proof (with any k) I see that my proof method is the same Sheer Shearing method.Therefore, we can see all facts in this topic as:- Generalization of proof 69 or- Interesting area property of orthodiagonal inscribed quadrilateral with proof using Sheer Shearing method.Best regards,Bui Quang Tuan

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