Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.7 AIME, harder Find the answer

Example 1. $P$ is a point inside an equilateral $\triangle A B C$ , and the ratio of the sizes of $\angle A P B, \angle B P C, \angle C P A$ is $5: 6: 7$ . Then the ratio of the sizes of the three interior angles of the triangle with sides $P A, P B, P C$ is ( ) (from smallest to largest).

Pick one

Solution

From the given, we know that $\angle A P B=100^{\circ}, \angle B P C=$ $120^{\circ}, \angle C P A=140^{\circ}$ .

From the proof of the previous problem, we know that the three interior angles of the triangle with sides of lengths $P A, P B, P C$ are (note the comparison and observation of the angles at one point):
$100^{\circ}-60^{\circ}, 120^{\circ}-60^{\circ}, 140^{\circ}-60^{\circ},$

which are $40^{\circ}, 60^{\circ}, 80^{\circ}$ , respectively. Therefore, their ratio is $2: 3: 4$ , so the answer is A.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.