A NEW PROOF OF PTOLEMY S THEOREM

Transcription

1 A NEW PROOF OF PTOLEMY S THEOREM DASARI NAGA VIJAY KRISHNA Abstract In this article we give a new proof of well-known Ptolemy s Theorem of a Cyclic Quadrilaterals 1 Introduction In the Euclidean geometry, Ptolemy s Theorem is a relation between the four sides two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle) The Theorem is named after the Greek astronomer mathematician Ptolemy (Claudius Ptolemaeus) Ptolemy used the Theorem as an aid in creating his table of chords, a trigonometric table that he applied to astronomy If the cyclic quadrilateral is given with its four vertices A, B, C, D in order, then the Theorem states that BD AB CD + BC AD This relation may be verbally expressed as follows: If a quadrilateral is inscribed in a circle then the product of the measures of its diagonals is equal to the sum of the products of the measures of the pairs of opposite sides Moreover, the converse of the Ptolemy s Theorem is also true: If the sum of the products of two pairs of opposite sides of a quadrilateral is equal to the product of its diagonals, then the quadrilateral can be inscribed in a circle In this short paper we deal with the new proof for this celebrated Theorem Unfortunately or fortunately what ever the proofs are available in the literature (some of them can be found in [5], [1], [3] [6]) are just based on constructing some particular lines applying similarity using little angle chasing between the triangles thus formed But in our present proof which is quite different from the available proofs, we won t construct any line In fact we will just try to prove above mentioned result by using the simple consequence of Theorem obtained by the Stewart Theorem on the diagonals of convex quadrilateral Our proof actually follows immediately from Equation (6) of Theorem In the end of the article we will also prove two characterizations of a Bicentric Quadrilateral Main Theorems Theorem 1 Let P be the point of intersection of diagonals BD of a convex quadrilateral ABCD M is an arbitrary point in the plane then (1) CP AM DP BD BM + AP CM BP BD DM AP P C BP P D, 1

2 DASARI NAGA VIJAY KRISHNA () Area(BCD) AM Area(D) BM + Area(ABD) CM Area(ABC) DM Area(ABC) (AP P C BP DP ) A M B P D C Fig 1 Proof Since P M is a cevian for triangles AMC BMD by Stewart s Theorem we have P M CP AM + AP CM AP P C P M DP BD BM + BP BD DM BP P D By setting the right sides of these two equations equal to each other, we obtain (1) Using the property that a cevian divides a triangle into two triangles whose ratio between areas is equal to the ratio between corresponding bases we have AP Area(AP D) Area(AP B) Area(AP D) + Area(AP B) Area(D) Area(B) Area(D) + Area(B) Area(ABD) Area(ABCD) In the same manner, CP Area(CBD) Area(ABCD), BP BD By replacing these ratios in (1), we get () Area(B) Area(ABCD), DP BD Area(D) Area(ABCD) Theorem Let P be the point of intersection of diagonals BD of a cyclic quadrilateral ABCD M is an arbitrary point in the plane then (3) AP P C BP P D, (4) CP AM + AP CM DP BD BM + BP BD DM, (5) Area(BCD) AM +Area(ABD) CM Area(D) BM +Area(ABC) DM, (6) BD BC CD AM + AB AD CM AD CD BM + AB BC DM

3 A NEW PROOF OF PTOLEMY S THEOREM 3 A B P D Proof From (1), for any point M we have Fig C AP P C BP P D CP AM DP BD BM + AP CM BP BD DM Let O be the circumcenter of the quadrilateral ABCD Taking M as O, we have AO BO CO DO R AP P C BP P D CP AO DP BD BO + AP CO BP BD DO ( ) AP + CP R BP + DP 0 BD Combining (3) with (1) () we get (4) (5) Now for (6), we will follow the well known fact that area of a triangle whose sides are a, b c circumradius R equals abc We have Area(BCD) AM + Area(ABD) CM Area(D) BM + Area(ABC) DM, then (7) BC BD CD AM AB BD AD + CM AD CD BM + AB BC DM, BD (BC CD AM + AB AD CM ) (AD CD BM + AB BC DM ) 3 Main Results 31 Ptolemy s Theorems In this section we present a new proof of the famous Ptolemy s Theorem Theorem 31 (Ptolemy s Theorem) For any cyclic quadrilateral ABCD with diagonals BD holds BD AB CD + BC AD

4 4 DASARI NAGA VIJAY KRISHNA Proof From Equation (6) we have BD BC CD AM + AB AD CM AD CD BM + AB BC DM Since it is true for any point M, let us take M as A So BD BC CD AA + AB AD CA AD CD BA + AB BC DA AB CD + AD BC, BD AB CD + BC AD Now we prove Ptolemy s Second Theorem Theorem 3 (Ptolemy s Second Theorem) For any cyclic quadrilateral ABCD with diagonals BD holds BC CD + AB AD BD AD CD + AB BC Proof From Equation (6) we have BD BC CD AM + AB AD CM AD CD BM + AB BC DM Let O be the circumcenter of the quadrilateral ABCD R AO BO CO DO Taking M as O we have BD BC CD R + AB AD R AD CD R + AB BC R, BC CD + AB AD BD AD CD + AB BC 3 Characterizations of bicentric quadrilaterals Now we present the proof of two Theorems about bicentric quadrilaterals Let us recall here the definition (for more details see for example [7]) Definition A bicentric quadrilateral is a convex quadrilateral that has both an incircle a circumcircle Theorem 33 Let ABCD be any bicentric quadrilateral with diagonals BD If I is incenter of ABCD then AI CI BI DI BC CD + AB AD BD AD CD + AB BC Proof Since ABCD is a cyclic quadrilateral, hence by Ptolemy s Second Theorem we have BC CD + AB AD (8) BD AD CD + AB BC Also from the equation (6) we have BD BC CD AM + AB AD CM AD CD BM + AB BC DM

5 A NEW PROOF OF PTOLEMY S THEOREM 5 Taking M as I we have (9) BD BC CD AI + AB AD CI AD CD BI + AB BC DI Using triangle s area formula, we obtain ( A Area(AIB) r AB AI BI sin + B ), where r is the inradius of the quadrilateral ABCD In the same manner, ( B Area(BIC) r BC BI CI sin + C ), ( C Area(CID) r CD CI DI sin + D ), ( D Area(DIA) r AD DI AI sin + A Since ABCD is a cyclic quadrilateral, we have ( A sin + B ) ( C sin + D ) ( B sin + C ) ( D sin + A ) Then AI BI CI DI AB CD, BI CI AI DI BC AD Dividing these two equation on each other, we obtain ( ) AI AB AD CI CB CD, ( ) BI BA BC DI DA DC Now using Equations (8) (9), we notice that So BD BC CD AI + AB AD CI BC CD + AB AD AD CD BI + AB BC DI AD CD + AB BC By multiplying these, we get AD CI BD BC DI, BC AI BD AD DI AI CI BI DI BD )

6 6 DASARI NAGA VIJAY KRISHNA Hence it follows the required result Theorem 34 Let ABCD be any bicentric quadrilateral with diagonals BD If t a, t b, t c, t d be the lengths of the tangents to its incircle from the vertices A, B, C, D respectively then BD t a + t c t b + t d Proof From Equation (6) we have BD BC CD AM + AB AD CM AD CD BM + AB BC DM Let I be the incenter of quadrilateral ABCD r be its inradius Taking M as I, we have (10) BD BC CD AI + AB AD CI AD CD BI + AB BC DI Since AI sin( r A ), BI sin( r B ), CI DI r r sin( C ) sin( D ) r t at b t c + t b t c t d + +t c t d t a + t d t a t b t a + t b + t c + t d By replacing AI, BI, CI, DI with their equivalent expressions in terms of t a, t b, t c, t d (can be found in [], [4] ), AB t a + t b, BC t b + t c, CD t c + t d DA t d + t a in (10) by some computation, the required result follows In order to prove the same result in other way, we deal with the following proof Another proof Let m : cot ( ) ( A n : cot B ) Observe that t a r m t b r n We have ( ) ( ) t c C A r cot tan 1 m, hence t c r m similarly t d r n So (11) BC CD AI (t a + t b ) (t c + t d ) ( r + t a) ( r4 (1 + mn) (m + n) m + 1 ) AB AD CI mn m ( AD CD BI r4 (1 + mn) (m + n) n + 1 ) AB BC DI mn n Now we have ( ) m + 1 BD ( m) n + 1 t a + t c t n b + t d Hence the result is alternatively proved 4 Acknowledgement support In completing this article, the author highly appreciates the referee s contribution for his valuable comments suggestions

7 REFERENCES 7 References [1] G I S Amarasinghe A concise elementary proof for the ptolemy s theorem Global Journal of Advanced Research on Classical Modern Geometries, (1):0 5, 013 [] M Josefsson Calculations concerning the tangent lengths ad tangency chords of a tangential quadrilateral Forum Geometricorum, 10: , 010 [3] M Josefsson Characterizations of bicentric quadrilaterals Forum Geometricorum, 10: , 010 [4] S H Kung Proof without words: the law of cosines via ptolemy s theorem Mathematics Magazine, 199 [5] E J N Schaumberger A vector approach to ptolemy s theorem Mathematics Magazin, 77(5):1, 004 [6] O Shisha On ptolemy s theorem International Journal of Mathematics Mathematical Sciences, 14():410, 1991 [7] Wikipedia Bicentric quadrilateral url: https : / / en wikipedia org / wiki / Bicentric_quadrilateral Department of Mathematics, Narayana Educational Institutions, Machilipatnam, Bangalore, India address: vijay @gmailcom