HMMT 十一月 2021 · THM 赛 · 第 1 题

HMMT November 2021 — THM Round — Problem 1

Let n be the answer to this problem. In acute triangle ABC , point D is located on side BC so that ∠ BAD = ∠ DAC and point E is located on AC so that BE ⊥ AC. Segments BE and AD intersect ◦ ◦ at X such that ∠ BXD = n . Given that ∠ XBA = 16 , find the measure of ∠ BCA.

解析

Let n be the answer to this problem. In acute triangle ABC , point D is located on side BC so that ∠ BAD = ∠ DAC and point E is located on AC so that BE ⊥ AC. Segments BE and AD intersect ◦ ◦ at X such that ∠ BXD = n . Given that ∠ XBA = 16 , find the measure of ∠ BCA. Proposed by: Joseph Heerens Answer: 53 Solution: B ◦ 16 D ◦ n X ◦ n C A E ◦ ◦ ◦ Since BE ⊥ AC , ∠ BAE = 90 − ∠ ABE = 74 . Now, n = 180 − ∠ BXA = ∠ EBA + ∠ BAD = ◦ 74 ◦ ◦ 16 + = 53 . 2