PUMaC 2014 · 团队赛 · 第 11 题

PUMaC 2014 — Team Round — Problem 11

[ 8 ] 4 ABC has AB = 4 and AC = 6. Let point D be on line AB so that A is between B and D . Let the angle bisector of \ BAC intersect line BC at E , and 2 let the angle bisector of DAC intersect line BC at F . Given that AE = AF , find the square of the circumcircle’s radius’ length.

解析

[ 8 ] 4 ABC has AB = 4 and AC = 6. Let point D be on line AB so that A is between B and D . Let the angle bisector of ∠ BAC intersect line BC at E , and let the angle bisector of DAC intersect line BC at F . Given that AE = AF , find the square of the circumcircle’s radius’ length. 3 Solution: AB AC Let the length of the circumradius be R . Sine law states that = = 2 R . So, sin C sin B 2 2 2 2 2 AB + AC = 4 R (sin B + sin C ). Let H be the intersection point that results when drawing an altitude from A to line BC . Then we get: ◦ ∠ BAH = ∠ EAH ± ∠ BAE = 45 ± ∠ BAE WLOG let the above satisfy minus instead of plus (which means the analogous equation for ∠ CAH satisfies plus instead of minus). Then, we also get: ◦ ∠ ACB = ∠ AEB − ∠ CAE = 45 − ∠ CAE Since ∠ BAE = ∠ CAE , we see that ∠ BAH = ∠ ACB . So, 4 BAH is similar to 4 ACH . A 16+36 2 little more work will show that sinB = cosC , so R = = 13 . 4