MathDB

3

Part of 2009 Sharygin Geometry Olympiad

Problems(4)

isosceles trapezoid

Source: I.F.Sharygin contest 2009 - Correspondence round - Problem 3

5/31/2009
The bisectors of trapezoid's angles form a quadrilateral with perpendicular diagonals. Prove that this trapezoid is isosceles.
geometrytrapezoidparallelogramrectangleincentergeometry proposed
segment of projections bisects segment of feet of altitudes

Source: 2009 Sharygin Geometry Olympiad Final Round problem 3 grade 8

7/26/2018
Let AHaAH_a and BHbBH_b be the altitudes of triangle ABCABC. Points PP and QQ are the projections of HaH_a to ABAB and ACAC. Prove that line PQPQ bisects segment HaHbH_aH_b.
(A.Akopjan, K.Savenkov)
geometryprojectionaltitudes
circumcircle, incircles, and concurrent lines

Source: 2009 Sharygin Geometry Olympiad Final Round problem 3 grade 9

7/26/2018
Quadrilateral ABCDABCD is circumscribed, rays BABA and CDCD intersect in point EE, rays BCBC and ADAD intersect in point FF. The incircle of the triangle formed by lines AB,CDAB, CD and the bisector of angle BB, touches ABAB in point KK, and the incircle of the triangle formed by lines AD,BCAD, BC and the bisector of angle BB, touches BCBC in point LL. Prove that lines KL,ACKL, AC and EFEF concur.
(I.Bogdanov)
geometrycircumcircleincircleconcurrencyconcurrent
projection of intersection of external <C-bisector and AB, on OI

Source: 2009 Sharygin Geometry Olympiad Final Round problem 3 grade 10

7/26/2018
The cirumradius and the inradius of triangle ABCABC are equal to RR and r,O,Ir, O, I are the centers of respective circles. External bisector of angle CC intersect ABAB in point PP. Point QQ is the projection of PP to line OIOI. Find distance OQ.OQ.
(A.Zaslavsky, A.Akopjan)
geometrydistanceangle bisector