MathDB

12

Part of 2023 AIME

Problems(2)

Miquel Point

Source: 2023 AIME I #12

2/8/2023
Let △ABC\triangle ABC be an equilateral triangle with side length 5555. Points DD, EE, and FF lie on sides BC‾\overline{BC}, CA‾\overline{CA}, and AB‾\overline{AB}, respectively, with BD=7BD=7, CE=30CE=30, and AF=40AF=40. A unique point PP inside △ABC\triangle ABC has the property that ∡AEP=∡BFP=∡CDP.\measuredangle AEP=\measuredangle BFP=\measuredangle CDP. Find tan⁡2(∡AEP)\tan^{2}\left(\measuredangle AEP\right).
geometry
why maa why

Source: 2023 AIME II / 12

2/16/2023
In △ABC\triangle ABC with side lengths AB=13AB=13, BC=14BC=14, and CA=15CA=15, let MM be the midpoint of BC‾\overline{BC}. Let PP be the point on the circumcircle of △ABC\triangle ABC such that MM is on AP‾\overline{AP}. There exists a unique point QQ on segment AM‾\overline{AM} such that ∠PBQ=∠PCQ\angle PBQ = \angle PCQ. Then AQAQ can be written as mn\frac{m}{\sqrt{n}}, where mm and nn are relatively prime positive integers. Find m+nm+n.
AMCAIMEAIME II