Triangles and parallelograms of equal area in an ellipse

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1 1 Tringles nd prllelogrms of equl re in n ellipse Roert Buonpstore nd Thoms J Osler Mthemtics Deprtment RownUniversity Glssoro, NJ 0808 USA Introduction In the pper [1], Euler looked t certin properties of the conic sections nd tried to find other curves tht shred these properties In the eighteenth century, mthemticins were fmilir with mny properties of the prol, ellipse nd hyperol tht hve een neglected in our modern eduction This pper is out one such ignored property of the ellipse which we rediscovered in order to understnd Euler s work We will study n interesting fmily of prllelogrms inscried in the ellipse, ll of which hve the sme re We egin y defining few new terms, dimeters, reciprocl dimeters nd reciprocl points in n ellipse Figure 1

2 A dimeter of n ellipse is ny chord tht psses through the center In Figure 1 MM nd mm re dimeters Now strt with ny dimeter MM We sy tht the dimeter mm is reciprocl to dimeter MM if it is prllel to the tngent line to the ellipse t M If we strted with dimeter mm, then MM would e the reciprocl dimeter We sy tht the points m nd m re reciprocl to the point M Euler ssumes tht his reders were fmilir with reciprocl dimeters nd points He lso ssumed tht his reders would e wre tht the re of the prllelogrm MmM m is constnt, regrdless of the choice of the initil dimeter, nd is equl to The re of the tringles CMm nd CMm re lso constnt nd equl / Before we derive our min result, we review prmetric equtions for the ellipse nd their geometric consequences Prmetric equtions for the ellipse These reciprocl dimeters hve n interesting reltion to the prmetric form of the eqution of the ellipse given y the equtions (1) x = cosθ nd y = sinθ Figure

3 3 In Figure we see two circles centered t point C with rdii nd The ry CRS mkes ngle θ with the x-xis nd intersects the smller circle t R nd the lrger circle t S From R extend horizontl line nd from S drop verticl line These two lines intersect t the point M This point M is on the ellipse given y the prmetric equtions (1) As the ngle θ vries etween 0 nd π, the point M genertes the entire ellipse Notice tht the ry CM which identifies the point ( x, y) on the perimeter of the ellipse differs from the ry CS tht is mde y the prmeter θ The ngle θ is known historiclly s the eccentric nomly It is importnt to note tht this is not the usul polr ngle ssocite with the point ( x, y) Reciprocl dimeters nd the eccentric nomly Figure 3 To see the reltion etween the reciprocl dimeters nd the eccentric nomly, consider Figure 3 Strt with the rdius CR mking eccentric nomly θ, to identify the point on the ellipse M Now increse the eccentric nomly y π / to identify the rdius ry CS nd corresponding point on the ellipse m We will show tht this point m is

4 4 reciprocl to M Thus reciprocl points on the ellipse hve their relted eccentric nomlies seprted y the ngle π / get To see tht this is true, we use (1) to clculte the slope of the tngent t M We dy dx cosθ dθ = = cotθ sinθ dθ Therefore the slope of the reciprocl ry Cm is given y slope of the ry CR defined y the eccentric nomly is cotθ (Notice tht when the tn θ, then the slope of the corresponding ry CM defined y the point on the ellipse is lwys given y Thus the slope of the ry CS is given y tnθ ) cotθ But the identity cotθ = tn( θ + π / ) demonstrtes the truth of the reltion etween M nd m just stted Thus the coordintes of the point m reciprocl to M re given y () t = cos( θ + π / ) = sinθ nd u = sin( θ + π / ) = cosθ The re of the tringle CMm is constnt Figure 4 y + u The re of the qudrilterl ABMm shown in Figure 4 is ( x t) Sutrcting the res of tringles ACm nd CBM we get the re of the tringle CMm y + u ( t) u xy Are CMm = ( x t)

5 5 This simplifies to (1) nd () we get Are CMm = xu ty Sustituting the vlues of these vriles in terms of θ from cos θ + sin θ = This proves tht the re of the tringle CMm is constnt In the sme wy the re of tringle CMm is nd thus the re of the prllelogrm MmM m is This completes our study of the tringles nd prllelogrms of equl re in the ellipse Reference [1] Greve, Edwrd nd Osler, Thoms J, Trnsltion with notes of Euler's pper E83, On some properties of conic sections tht re shred with infinitely mny other curved lines On the we t the Euler Archive

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