MathDB

5

Part of 2019 Sharygin Geometry Olympiad

Problems(4)

Sharygin CR 2019 P5

Source:

3/6/2019
Let A,B,CA, B, C and DD be four points in general position, and ω\omega be a circle passing through BB and CC. A point PP moves along ω\omega. Let QQ be the common point of circles ⊙(ABP)\odot (ABP) and ⊙(PCD)\odot (PCD) distinct from PP. Find the locus of points QQ.
geometry
Triangle with an angle

Source: Sharygin Finals 2019 Day2 Grade 8 P5

7/31/2019
Grade8 P5 of Sharygin 2019 Finals
Circumradius of a cyclic quadrilateral

Source: Sharygin 2019 Finals Day2 Grade 9 P1

7/31/2019
Let RR be the circumradius of a circumscribed quadrilateral ABCDABCD . Let h1h_1 and h2h_2 be the altitudes from AA to BCBC and CDCD respectively. Similarly h3h_3 and h4h_4 are the altitudes from CC to ABAB and ADAD. Prove that h1+h2−2Rh1h2=h3+h4−2Rh3h4\frac {h_1+h_2- 2R}{h_1h_2}=\frac {h_3+h_4-2R}{h_3h_4}
concurrency wanted, feet of altitudes, intersection with circumcircle related

Source: Sharygin 2019 final 10 P5

8/15/2019
Let AA1,BB1,CC1AA_1, BB_1, CC_1 be the altitudes of triangle ABCABC, and A0,C0A0, C0 be the common points of the circumcircle of triangle A1BC1A_1BC_1 with the lines A1B1A_1B_1 and C1B1C_1B_1 respectively. Prove that AA0AA_0 and CC0CC_0 meet on the median of ABC or are parallel to it
geometrycircumcircleconcurrencyaltitudes