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Previous Puzzles of the Month + Solutions

September-October 2008, Puzzle nr 119
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Puzzle # 119 Difficulty level: , general math knowledge.

 Perpendicular or not?   The large circle (O1, B) above circumscribes 2 smaller circles (with centers O2 and O3, respectively) and an isosceles triangle, whose base stands on the diameter of the large circle. As shown in the drawing, each geometric shape touches the 3 other ones. Points A, O1, O2 and B are aligned. Demonstrate that the segment O3A is perpendicular to the diameter AB. Keywords: incircles, isosceles, perpendicular, secant-tangent. Related puzzles: - Ratio of similar triangles. - Soccer ball problem. - Ad angolo retto? Il grande cerchio (O1, B) circoscrive 2 cerchi minori (con i rispettivi centri O2 e O3) e un triangolo isoscele, la cui base si confonde col diametro del grande cerchio. Ogni figura geometrica tocca ognuna delle altre tre. I punti A, O1, O2 e B sono allineati. Dimostrare che il segmento O3A è perpendicolare al diametro AB. Parole chiave: cerchio inscritto, isoscelo, perpendicolare. Suggerisci un'altra soluzione Chiudi - A angle droit? Le grand cercle (O1, B) circonscrit 2 autres cercles plus petits (ayant pour centres O2 et O3, respectivement) ainsi qu’un triangle isocèle, dont la base se trouve sur le diamètre du grand cercle. Chaque figure géométrique touche les 3 autres. Les points A, O1, O2 et B sont alignés. Démontrer que le segment O3A est bien perpendiculaire au diamètre AB. Mots clés: cercle inscrit, isocèle, perpendiculaire. Propose une autre solution Fermer Source of the puzzle: Sangaku tablet of Gunma prefecture (群馬県; Gunma-ken), 1803. ©G. Sarcone.

 More Math Facts behind the puzzle Secant-tangent theorem: fig. 1) c2 = a(a + b) and fig. 2) c2 = a(a + 2r) The angle, formed between a tangent line and a chord, is congruent to the inscribed angle on the other side of the chord and subtended by the chord (angle α' = angle α, see fig. 3). Then, with similar triangles: c / (a + b) = a / c

Previous puzzles of the month...
 July-Aug 08: ratio of triangles May-June 08: geometry of the bees Febr-March 08: parrot sequence... Dec 07-Jan 08: probable birthdates? Oct-Nov 2007: infinite beetle path Aug-Sept 07: indecisive triangle June-July 07: Achtung Minen! April-May 07: soccer balls Febr-March 07: prof Gibbus' angle Jan 07: triangles to square Aug-Sept 2006: balance problem June-July 06: squared strip Apr-May 06: intriguing probabilities Febr-March 06: cows & chickens Dec 05-Jan 06: red monad Sept-Oct 2005: magic star July-Aug 05: cheese! May 05: stairs to square Febr 05: same pieces Sept-Oct 2004: odd triangles June 04: pizza's pitfalls May 04: Pacioli puzzle... Puzzle Archive More Puzzles

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