MathDB

G

Part of 2023 Taiwan TST Round 1

Problems(2)

A line that bisects two angles

Source: 2023 Taiwan TST Round 1 Independent Study 2-G

3/16/2023
Let Ω\Omega be the circumcircle of an isosceles trapezoid ABCDABCD, in which ADAD is parallel to BCBC. Let XX be the reflection point of DD with respect to BCBC. Point QQ is on the arc BCBC of Ω\Omega that does not contain AA. Let PP be the intersection of DQDQ and BCBC. A point EE satisfies that EQEQ is parallel to PXPX, and EQEQ bisects ∠BEC\angle BEC. Prove that EQEQ also bisects ∠AEP\angle AEP.
Proposed by Li4.
geometrycircumcircletrapezoidgeometric transformationreflection
Lines connecting centroids

Source: 2023 Taiwan TST Round 1 Independent Study 1-G

3/15/2023
Let ABCABC be a triangle. Let ABC1,BCA1,CAB1ABC_1, BCA_1, CAB_1 be three equilateral triangles that do not overlap with ABCABC. Let PP be the intersection of the circumcircles of triangle ABC1ABC_1 and CAB1CAB_1. Let QQ be the point on the circumcircle of triangle CAB1CAB_1 so that PQPQ is parallel to BA1BA_1. Let RR be the point on the circumcircle of triangle ABC1ABC_1 so that PRPR is parallel to CA1CA_1.
Show that the line connecting the centroid of triangle ABCABC and the centroid of triangle PQRPQR is parallel to BCBC.
Proposed by usjl
Taiwangeometry