MathDB

8

Part of 2023 Caucasus Mathematical Olympiad

Problems(2)

Prove that the triangle $A^{'}B^{'}C^{'}$ is equilateral.

Source: Caucasus MO 2023

7/16/2023
Let ABCABC be an equilateral triangle with the side length equals a+b+ca+ b+ c. On the side ABAB{} of the triangle ABCABC points C1C_1 and C2C_2 are chosen, on the side BCBC points A1A_1 and A2A_2, arc chosen, and on the side CACA points B1B_1 and B2B_2 are chosen such that A1A2=CB1=BC2=a,B1B2=AC1=CA2=b,C1C2=BA1=AB2=cA_1A_2 = CB_1 = BC_2 = a, B_1B_2 = AC_1 = CA_2 = b, C_1C_2 = BA_1 = AB_2 = c. Let the point A’A^{’} be such that the triangle A′B2C1A^{'} B_2C_1 is equilateral, and the points AA and A′A^{'} lie on different sides of the line B2C1B_2C_1. Similarly, the points B’B^{’} and C′C^{'} are constructed (the triangle B′C2A1B^{'} C_2A_1 is equilateral, and the points BB and B’B^{’} lie on different sides of the line C2A1C_2A_1; the triangle C′A2B1C^{'} A_2B_1 is equilateral, and the points CC and C′C^{'} lie on different sides of the line A2B1A_2B_1). Prove that the triangle A′B′C′A^{'}B^{'}C^{'} is equilateral.
geometry
Cute geometry

Source: Caucasus MO 2023, senior, day 2, P8

3/12/2023
Let ABCABC be an acute-angled triangle, and let AA1,BB1,CC1AA_1, BB_1, CC_1 be its altitudes. Points A′,B′,C′A', B', C' are chosen on the segments AA1,BB1,CC1AA_1, BB_1, CC_1, respectively, so that ∠BA′C=∠AC′B=∠CB′A=90o\angle BA'C = \angle AC'B = \angle CB'A = 90^{o}. Let segments AC′AC' and CA′CA' intersect at B"B"; points A",C"A", C" are defined similarly. Prove that hexagon A′B"C′A"B′C"A'B"C'A"B'C" is circumscribed.
geometry