12 Geometry Exercises by the Romantics of Geometry

issue 46 , No. 541-552 edition 1st: 07-03-2017

edited by: Chronopoulos Takis (parmenides51) This file contains the 46th dozen of exercises that were proposed to the facebook group Romantics of Geometry. Under each problem, the link to where it was at first proposed (and usually got solved) is given. The problems usually contain the data that cannot be seen in figures. After sufficient time, the sources of the exercises shall be added. The lemmas that are created during someone’s solution, shall follow the numbering of each proposal in a separate page. Edition follows the Greek version of every pdf.

(figure of problem 515) https://web.facebook.com/groups/parmenides52

(Romantics of Geometry - Facebook group)

1

541 Seiichi Kirikami (original) Equilateral triangle and focus sharing hyperbolas Let ABC be a equilateral triangle and P a point. D, E, F = projections of P on BC, CA, AB respectively. We suppose that D, E, F are between B and C, C and A, A and B respectively. c = hyperbola with its foci B and C through D. d = hyperbola with its foci C and A through E. e = hyperbola with its foci A and B through F. c, d and e concur in points Q and R.

541 Link: https://www.facebook.com/groups/parmenides52/permalink/1239718126141891/

https://web.facebook.com/groups/parmenides52

(Romantics of Geometry - Facebook group)

2

542 Hatzipolakis Antreas (original) Let APBC' be aa quadrangle with PAC' = PBC' = 90 d. The perpendicular to AB at B intersects the perpendicular to AP at P at the point A2. A2C' is parallel to AP.

source: https://groups.yahoo.com/neo/groups/Hyacinthos/conversations/messages/25534

542 Link: https://www.facebook.com/groups/parmenides52/permalink/1240002129446824/

https://web.facebook.com/groups/parmenides52

(Romantics of Geometry - Facebook group)

3

543 Tran Quang Hung (original) Let A1A2...A17 is regular 17-gon. XYZ is Morley triangle of triangle A1A8A14. Prove that YZ is parallel to diagonal A6A15.

543 Link: https://www.facebook.com/groups/parmenides52/permalink/1240321916081512/

https://web.facebook.com/groups/parmenides52

(Romantics of Geometry - Facebook group)

4

544 Tran Quang Hung (original) Let A1A2..A7 is a regular heptangon. Let XYZ be Morley triangle of triangle A1A5A6 Then XY pass through center O of polygon and XY is parallel to diagonal A2A6.

544 Link: https://www.facebook.com/groups/parmenides52/permalink/1240353859411651/

https://web.facebook.com/groups/parmenides52

(Romantics of Geometry - Facebook group)

5

545 Tran Quang Hung (original) Flower of regular pentagon Reflect a regular pentagon through its sides we get a star regular pentagon whoes sides devides the sides of reference regular pentagon in the golden ratio.

545 Link: https://www.facebook.com/groups/parmenides52/permalink/1240378679409169/

https://web.facebook.com/groups/parmenides52

(Romantics of Geometry - Facebook group)

6

546 Tran Quang Hung (original) Taylor circle in O'clock Let 1,2,3....,12 is a regular dodecagon (O'clock). Then Taylor circle of triangle (4,8,11) passes through midpoint 8.5 of segment (8,9) and midpoint 8.75 of segment (8.5,9).

546 Link: https://www.facebook.com/groups/parmenides52/permalink/1240401762740194/

https://web.facebook.com/groups/parmenides52

(Romantics of Geometry - Facebook group)

7

547 Tran Quang Hung (original) Let A1A2...A17 is regular 17-gon. Then Simson line of A9 with respect to triangle A1A8A14 is perpendicular to side A15A16.

547 Link: https://www.facebook.com/groups/parmenides52/permalink/1241113142669056/

https://web.facebook.com/groups/parmenides52

(Romantics of Geometry - Facebook group)

8

548 Tran Quang Hung (original) Let ABC be a heptagon triangle with incenter I. DEF is cevian triangle of I. Then circumcircle (K) of DEF passes through A. (K) cuts BC again at P. Then both PI,DK bisect the segment EF.

548 Link: https://www.facebook.com/groups/parmenides52/permalink/1241145059332531/

https://web.facebook.com/groups/parmenides52

(Romantics of Geometry - Facebook group)

9

549 Kadir Altintas

549 Link: https://www.facebook.com/groups/parmenides52/permalink/1241498029297234/

https://web.facebook.com/groups/parmenides52

(Romantics of Geometry - Facebook group)

10

550 Makris Ioannis The big rectangle consists of 12 squares. The blue square has area 1 . Which is the area of the rectangle?

550 Link: https://www.facebook.com/groups/parmenides52/permalink/1241912819255755/

https://web.facebook.com/groups/parmenides52

(Romantics of Geometry - Facebook group)

11

551 Tran Quang Hung (original) Let ABC be a triangle, I is incenter. A'B'C' is circumcevian triangle of I. B'C' cuts AC,AB at Ac,Ab. Similarly, we have Ba,Bc,Cb,Ca. Consider circle (wa) center A passes through Ac,Ab. Similarly we have circle (wb),(wc). Circle (K) is tangent to (wa),(wb),(wc) externally. Circle (L) is tangent to (wa),(wb),(wc) internally. Then line KL is parallel to OI line of ABC.

551 Link: https://www.facebook.com/groups/parmenides52/permalink/1242242372556133/

https://web.facebook.com/groups/parmenides52

(Romantics of Geometry - Facebook group)

12

552 Kadir Altintas

552 Link: https://www.facebook.com/groups/parmenides52/permalink/1242240212556349/

https://web.facebook.com/groups/parmenides52

(Romantics of Geometry - Facebook group)

13

12 geometry exercises by Romantics of Geometry, issue 46, No 541 ...

problems usually contain the data that cannot be seen in figures. After sufficient time, the sources of. the exercises shall be added. The lemmas that are created during someone's solution, shall follow the. numbering of each proposal in a separate page. Edition follows the Greek version of every pdf. (figure of problem 515).
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