MathDB

3

Part of Mathley 2014-15

Problems(4)

no of isosceles triangles with angle >120 in a regular 2013-gon

Source: Mathley 2014.1 p3

8/20/2020
Given a regular 20132013-sided polygon, how many isosceles triangles are there whose vertices are vertices vertex of given polygon and haave an angle greater than 120o120^o?
Nguyen Tien Lam, High School for Natural Science,Hanoi National University.
combinatoricscombinatorial geometryisosceles
Mathley 2014.2 p3 by Tran Quang Hung

Source:

8/20/2020
In a triangle ABCABC, DD is the reflection of AA about the sideline BCBC. A circle (K)(K) with diameter ADAD meets DB,DCDB,DC at M,NM,N which are distinct from DD. Let E,FE,F be the midpoint of CA,ABCA,AB. The circumcircles of KEM,KFNKEM,KFN meet each other again at LL, distinct from KK. Let KLKL meets EFEF at XX; points Y,ZY,Z are defined in the same manner. Prove that three lines AX,BY,CZAX,BY,CZ are concurrent.
Tran Quang Hung, Dean of the Faculty of Science, Thanh Xuan, Hanoi.
geometryconcurrentreflection
AK,BC, t are parallel or concurrent , incircle related

Source: Mathley 2014.3 p3

8/18/2020
Let the incircle γ\gamma of triangle ABCABC be tangent to BA,BCBA, BC at D,ED, E, respectively. A tangent tt to γ\gamma , distinct from the sidelines, intersects the line ABAB at MM. If lines CM,DECM, DE meet atK K, prove that lines AK,BCAK,BC and tt are parallel or concurrent.
Michel Bataille , France
geometryparallelconcurrentincircle
orthocenter wanted, tangents of circumcircle of 2 projections related

Source: Mathley 2015 p3

8/18/2020
A point PP is interior to the triangle ABCABC such that AP⊥BCAP \perp BC. Let E,FE, F be the projections of CA,ABCA, AB. Suppose that the tangents at E,FE, F of the circumcircle of triangle AEFAEF meets at a point on BCBC. Prove that PP is the orthocenter of triangle ABCABC.
Do Thanh Son, High School of Natural Sciences, National University, Hanoi
circumcircleorthocenterprojectionsgame strategy