MathDB

Problem 3

Part of 2001 Slovenia National Olympiad

Problems(4)

P.t. dist. is independent of point (Slovenia National MO 2001 1st Grade P3)

Source:

4/7/2021
For an arbitrary point PP on a given segment ABAB, two isosceles right triangles APQAPQ and PBRPBR with the right angles at QQ and RR are constructed on the same side of the line ABAB. Prove that the distance from the midpoint MM of QRQR to the line ABAB does not depend on the choice of PP.
geometry
Prove triangle areas equal (Slovenia National MO 2001 2nd Grade P3)

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4/7/2021
Let EE and FF be points on the side ABAB of a rectangle ABCDABCD such that AE=EFAE = EF. The line through EE perpendicular to ABAB intersects the diagonal ACAC at GG, and the segments FDFD and BGBG intersect at HH. Prove that the areas of the triangles FBHFBH and GHDGHD are equal.
geometry
point lies on common chord (Slovenia National MO 2001 3rd Grade P3)

Source:

4/7/2021
A point DD is taken on the side BCBC of an acute-angled triangle ABCABC such that AB=ADAB = AD. Point EE on the altitude from CC of the triangle is such that the circle k1k_1 with center EE is tangent to the line ADAD at DD. Let k2k_2 be the circle through CC that is tangent to ABAB at BB. Prove that AA lies on the line determined by the common chord of k1k_1 and k2k_2.
geometry
angle computation in triangle (Slovenia National MO 2001 4th Grade P3)

Source:

4/7/2021
Let DD be the foot of the altitude from AA in a triangle ABCABC. The angle bisector at CC intersects ABAB at a point EE. Given that ∠CEA=π4\angle CEA=\frac\pi4, compute ∠EDB\angle EDB.
geometry