MathDB
Repeated Sequence of Triangles

Source: 2013 AIME I Problem 13

March 15, 2013
geometrytrigonometrysimilar trianglesnumber theoryrelatively primegeometric sequencetrig identities

Problem Statement

Triangle AB0C0AB_0C_0 has side lengths AB0=12AB_0 = 12, B0C0=17B_0C_0 = 17, and C0A=25C_0A = 25. For each positive integer nn, points BnB_n and CnC_n are located on ABn−1‾\overline{AB_{n-1}} and ACn−1‾\overline{AC_{n-1}}, respectively, creating three similar triangles △ABnCn∼△Bn−1CnCn−1∼△ABn−1Cn−1\triangle AB_nC_n \sim \triangle B_{n-1}C_nC_{n-1} \sim \triangle AB_{n-1}C_{n-1}. The area of the union of all triangles Bn−1CnBnB_{n-1}C_nB_n for n≥1n\geq1 can be expressed as pq\tfrac pq, where pp and qq are relatively prime positive integers. Find qq.
Back to ProblemsView on AoPS