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18 Education update ■ For Parents, Educators & Students ■ MAY/JUNE 2011The DEAN’S COLUMNLove and Loss on Mother’s DayFigure 3By Pola Rosen, Ed.D.All Triangles are IsoscelesCBS reporter Magee Hickeyrecently gave a poignant performanceof her mother’s favoriteBy Alfred Posamentier, Ph.D.With the return of the geometry course to the and Figure 4, where CG and GE meet songs in a Cabaret show in NewNew York State curriculum, we owe our students outside the and triangle, Figure but 4, where the perpendiculars meet outside York the City. triangle, For some but the of perpendicularsus, Mother’sthe pleasure of being enriched in the subject GD and GF meet CA and CB outside the triangle. Day is a time to celebrate, exchangell Triangles and not just present are with Isoscelesthe essentials that will the triangle.gifts and enjoy the warmth of achool of be Educationtested. Towards that end, the presentation ofshared dinner. For others, it’s a timeAll Triangles are Isosceles“geometric fallacies” offer a deeper insight intoto recall days past, when music,samentier,rse to the Newthe subject Dean,Yorkthat SchoolStatewill lead to ofcurriculum,a better Educationwe owe our students the pleasureunderstandingadvice and good times were sharedofnotthejustgeometryand present a greater withcourseappreciation. thetoessentialsthe NewthatYorkwillStatebecurriculum,tested. Towardswe owethatourend,students the pleasureand are sadly lost. #acies” George Pólya, one of the great mathematiciansd in theofferof subjecta deeperour time, and said, notinsight“Geometry just presentinto is the withsubjectscience the of essentialsthat willthatleadwilltobeatested.betterTowards that end,tion.of “geometric correct reasoning fallacies” on incorrect offer a figures.” deeper We insight into the subject that will lead to a betterd a greater will demonstrate appreciation. below that making conclusionsathematicians based on of “incorrect” our time, figures said, can “Geometry lead us to is the science of correctimpossible results. Even the statements of theonee willofdemonstratefallacies the great sound mathematiciansbelow that makingabsurd. However, students of ourconclusionswill time, said,based“Geometryon “incorrect”is the science of correctcorrectesults. find figures.”Even the demonstration theWestatementswill of proving demonstrateof something the fallaciesbelow that thatsoundmakingabsurd.conclusionsHowever,based on “incorrect” The pleasures of living on East End Avenueion is absurd to be either frustrating or enchanting,are one among the great real estate secrets inusoftodependingimpossibleproving something theresults.spin Eventhat isteachertheabsurdputsstatementsto beon it.ofeitherthe fallaciesfrustratingsoundorabsurd. However, Figure 4heManhattan. It is a calm and quiet area, closendteacherthe Nevertheless, demonstrationputs on it. Nevertheless,follow of each provingfollowstatement somethingeachof the thatstatementis Figure absurdof the4 to“proof”be either frustrating to parks, or and many are unaware of its uniquee.ndingIt restson “proof” theon something and spin see the if you teacherthat can Euclid detect puts the onin mistake. it.hisNevertheless,Elements wouldfollownot haveeachbeenstatement of the “proof” charms that deserve attention. Families seekingIt rests on something that Euclid in his Elements The “proof” of the fallacy can be done with any a family friendly neighborhood in Manhattan,andefinition.detect the mistake. It rests on something that Euclid in his Elements would not have beenwould not have been able to resolve because of a of these figures. Follow the “proof” on any (or care a great deal about the areas educational andecause lack of a of lack a definition. of a definition.all) of these figures.recreational opportunities for their children. East(a triangleThewithFallacy:threeAnyunequalscalene trianglesides)(aistriangleisoscelesGIVEN:(a triangleDABC is scalene.having twoEnd Avenue offers great schools, Carl Schurzwith three unequal sides) is isosceles (a triangle PROVE: AC = BC (or DABC is isosceles) Park, and several buildings, such as 180 East Endny scalene triangle (a triangle with three unequal sides) is isosceles (a triangle having twoeles, we having must two draw equal sides). a few auxiliary line segments. “Proof”: Draw Since the ∠ACG bisector ≅ ∠BCG of and right Ave, have great outdoor gardens, gyms, communityrooms and other amenities that young fami-To prove that scalene DABC is isosceles, we ∠CDG ≅ right ∠CFG, DCDG ≅ DCFG (SAA).alene of ΔABC must . From draw is isosceles, a their few auxiliary point we line of must segments. intersection, draw Draw a few G, Therefore auxiliary draw DG perpendiculars line = FG segments. and CD = tohe “proof” of the fallacy can be done with any of these figures. Follow the “proof” on any Draw CF. (or Since all) the of AG bisector these lies especially of enjoy. 180 East End Ave also hasigures. the bisector of ∠C and the perpendicular bisector = BG (a point the perpendicular bisector of a rooftop pool with a solarium, a state-of-the-artpendicular D and F, respectively. bisector of . From their point of intersection, G, draw perpendiculars toof AB . From their point of intersection, G, draw a line segment is equidistant from the endpoints gym, and a lounge with pool table.eting IVEN: them ΔABC perpendiculars at is points scalene. to D AC and and F, CB respectively. ,meeting them of the line segment) and ∠ADG and ∠BFG are Carl Schurz Park, always beautiful, has undergonea renaissance during Mayor Bloomberg’sur possibilities at points D and for F, the respectively. above description for right various angles, scalene DDAG ≅ DFBG triangles: (Hypotenuse-Leg.).It should be noted that there are four possibilities Therefore DA = FB.administration, and a visit to Gracie Mansion’sted side ROVE: that the ACfor there triangle. = BC above are (or ΔABC four description possibilities is isosceles)for various for scalene the above It then description follows that for AC = various BC (by addition scalene in triangles: new renovation is a delightful experience.triangles: Figure 1, where CG and GE meet Figures 1 - 3; and by subtraction in Figure 4). Resplendent with an a grove of cherry trees,Proof”: meet inside the triangle.inside Since the ∠ACG triangle. ≅ ∠BCG and right ∠CDG ≅ right ∠CFG, At this ΔCDG point you ≅ ΔCFG may be (SAA). somewhat Therefore disturbed, DG = tulips. pansies and hydrangeas, Carl Schurz ParkFG and CD = CF. Since AG = BG (a point on wondering the perpendicular where the bisector error was of committed a line segment that isa delightful place to bring children, who loveequidistant from the endpoints of the line segment) permitted and ∠ADG this and fallacy ∠BFG to are occur. right By angles, rigorous ΔDAG ≅ΔFBG (Hypotenuse-Leg.). Therefore DA = FB. construction, you will find a subtle error in theIt then follows that AC = BC (by addition in Figures figures: 1 - 3; and by subtraction in Figure 4).a. The point G must be outside the triangle.t this point you may be somewhat disturbed, wondering where b. When the error perpendiculars was committed meet that the permitted sides of thisallacy to occur. By rigorous construction, you will find a subtle the triangle, error in one the will figures: meet a side between thevertices, while the other will not.a. The point G must be outside the triangle.In general terms used by Euclid, this dilemmawould remain an enigma, since the concept ofb. When perpendiculars meet the sides of the triangle, one will meet a side between the vertices,betweenness was not defined in his Elements.while the other will not.In the following discussion we shall prove thaterrors exist in the fallacious proof above. Ourn general terms used by Euclid, this dilemma would remainproofanusesenigma,Euclideansincemethods,the conceptbutofassumesbetweennessaas not defined in his Elements. Figure In 1 the following discussion definition we shall of prove betweenness. that errors exist in the fallaciousroof above. Our proof uses Euclidean methods, but assumes Begin a definition by considering of betweenness. the circumcircle ofFigure 2, where CG and GE meet on AB;egin by considering the circumcircle of ΔABCDABC (See Figure 5).See Figure 5).Figure 1;he bisector of ∠ACB must contain the midpoint G, of;since ∠ACG and ∠BCG are congruent inscribed angles).he perpendicular bisector of must bisect , andherefore pass through G. Thus, the bisector of ∠ACB andhe perpendicular bisector of intersect outside theriangle at G. This eliminates the possibilities illustrated inigures 1 and 2.Figure 5The bisector of ∠ACB must contain2Figure 2the midpoint G, of AB (since ∠ACG andFigure 5∠BCG are congruent inscribed angles). TheFigure Figure 3, where 2 CG and GE meet outside perpendicular bisector of AB must bisect ABthe triangle, but the perpendiculars GD and GF , and therefore pass through G. Thus, the bisectorow considerfall oninscribedAC and CBquadrilateral;ACBG. Since theof ∠ACBoppositeandanglesthe perpendicularof an inscribedbisector of(orABcyclic)uadrilateral are supplementary, m∠CAG + m∠CBG = 180°. intersect If ∠CAG outside the and triangle ∠CBG at are G. This right eliminates angles, thenwould be a diameter and ΔABC would be isosceles. the Therefore, possibilities since illustrated ΔABC in is Figures scalene, 1 and ∠CAG 2. andCBG are not right angles. In this case, one must be acute Now and the consider other inscribed obtuse. Suppose quadrilateral ∠CBG ACBG. is acutend ∠CAG is obtuse. Then in ΔCBG the altitude on must Since be the inside opposite the triangle, angles of while an inscribed obtuse ΔCAG, (orhe altitude on must be outside the triangle. (This is usuallycyclic) quadrilateralreadily acceptedare supplementary,without proof,m∠CAGbut can be+ m∠CBG = 180°. If ∠CAG and ∠CBG areasily proved.) The fact that one and only one of the perpendiculars intersects a side of the triangle betweenright angles, then CG would be a diameterhe vertices destroys the fallacious “proof.”and ∠ABC would be isosceles. Therefore, sinceFigure 3 ΔABC is scalene, ∠CAG and ∠CBG are not rightangles. In this case, one must be acute and theother obtuse. Suppose ∠CBG is acute and ∠CAGcontinued on page 27Figure 3, where meet outside the triangle, but the perpendiculars fall on;Figure 3igure 4, where meet outside the triangle, but the perpendicularsoutside the triangle.Family Livingon the Upper East Sidethe playground and a big dog and a small dog run.Living close to the Asphalt Green, the 91 stStreet ferry stop , the Vinegar Factory, and manywonderful local stores are also benefits. Theexcellent nearby community public schools suchas PS 290 and PS158 and private schools such asChapin and Brearley and others on the upper EastSide make the East End Avenue location a veryattractive neighborhood for families.East End Avenue is a relatively affordable andwell priced place to live with a wide range offamily sized apartments in fine buildings. Sinceboth parents often have careers and want to liveand work near their children, they can do thisand still have their families enjoy nature throughbeing close to Carl Shurz Park, and having a garden,terrace and or roof top terrace. Imagine myown pleasure in having a child of mine who grewup here, purchasing an apartment in the neighborhood,and the joy of being with my granddaughterin the playground at Carl Shurz that her dadloved so much as a child!Sloane Square Residential Real EstateMarsha FrancesExecutive Vice PresidentTel: (212) 327-3112 Cell: (917) 714-3009Fax: (212) 327-2069 Mfrances@SloaneSquareNYC.comSloaneSquareNYC.com1015 Madison Avenue, Suite 301New York, NY 10075180 East End Avenue - Sparkling Mint 3 BedroomThis is a stunning, high floor 3 bedroom, 3 bath plus study / 4th bedroom and sunny corneredbalcony with a beautiful mint renovation and incredible sweeping river views throughout! There isnow a contemporary open floor plan for the gallery, living room, dining room and kitchen for modernliving. The great north-east corner master bedroom suite with dressing area has well designed builtinsand limestone bath. Two other master bedrooms with built-ins and bath including an additionalstudy / 4th bedroom which has access to 3rd travertine and marble guest bath, top-of-the-line kitchenand dining counter complete the space. Included is all new electrical and plumbing, new windows,new HVACs, new kitchen appliances and bath fixtures, Viking and Sub-Zero kitchen appliances, porcelainfarm house sink, copper countertops, Anne Sacks tile, custom woodwork throughout includingmolding and wainscoting, wide plank pine floors, remote control blinds, oil rubbed bronze hardware.180 East End Avenue offers a rooftop pool and sun room, large beautifully landscape garden, state-ofthe-artgym, community room with pool table, bin in basement and garage ($265/monthly). ElectricitySurcharge of $540/month. Top-of-the-line apartment in top-of-the-line fullservice building. It is a must see!180 East End Avenue - ELEGANT ESTATEElegant spacious 5 room plus balcony with beautiful direct park and river views from every room.Enter into a grand gallery leading into an oversized living with balcony, huge formal dining room(could be third bedroom), windowed eat-in-kitchen, 2 large master bedroom suites with baths,powder room and lots of closet space throughout. Flexible layout works well for your own needsand design. 180 East End Avenue is a top of the line building with excellent full services and a terrificstaff. Building has a new very special landscaped garden, rooftop pool and sunroom, state-of-the-artgym, lounge with pool table, garage ($265/month), individual bins in basement. Energy surcharge is$399/month. It’s a great opportunity.