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Generalization of a Midpoint Theorem

Source: Indonesian Stage 1 TST for IMO 2022, Test 2 (Geometry)

December 23, 2021
Trianglegeometrymidpointconcurrence

Problem Statement

Given that ABCABC is a triangle, points Ai,Bi,Ci(i∈{1,2,3})A_i, B_i, C_i \hspace{0.15cm} (i \in \{1,2,3\}) and OA,OB,OCO_A, O_B, O_C satisfy the following criteria:
a) ABB1A2,BCC1B2,CAA1C2ABB_1A_2, BCC_1B_2, CAA_1C_2 are rectangles not containing any interior points of the triangle ABCABC, b) ABBB1=BCCC1=CAAA1\displaystyle \frac{AB}{BB_1} = \frac{BC}{CC_1} = \frac{CA}{AA_1}, c) AA1A3A2,BB1B3B2,CC1C3C2AA_1A_3A_2, BB_1B_3B_2, CC_1C_3C_2 are parallelograms, and d) OAO_A is the centroid of rectangle BCC1B2BCC_1B_2, OBO_B is the centroid of rectangle CAA1C2CAA_1C_2, and OCO_C is the centroid of rectangle ABB1A2ABB_1A_2.
Prove that A3OA,B3OB,A_3O_A, B_3O_B, and C3OCC_3O_C concur at a point.
Proposed by Farras Mohammad Hibban Faddila
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