MathDB

11

Part of 2019 AIME Problems

Problems(2)

so what's an excircle?

Source: 2019 AIME I #11

3/14/2019
In △ABC\triangle ABC, the sides have integers lengths and AB=ACAB=AC. Circle ω\omega has its center at the incenter of △ABC\triangle ABC. An excircle of △ABC\triangle ABC is a circle in the exterior of △ABC\triangle ABC that is tangent to one side of the triangle and tangent to the extensions of the other two sides. Suppose that the excircle tangent to BC‾\overline{BC} is internally tangent to ω\omega, and the other two excircles are both externally tangent to ω\omega. Find the minimum possible value of the perimeter of △ABC\triangle ABC.
excirclegeometry2019 AIME I
angle chase or bash

Source: 2019 AIME II #11

3/22/2019
Triangle ABCABC has side lengths AB=7,BC=8,AB=7, BC=8, and CA=9.CA=9. Circle ω1\omega_1 passes through BB and is tangent to line ACAC at A.A. Circle ω2\omega_2 passes through CC and is tangent to line ABAB at A.A. Let KK be the intersection of circles ω1\omega_1 and ω2\omega_2 not equal to A.A. Then AK=mn,AK=\tfrac{m}{n}, where mm and nn are relatively prime positive integers. Find m+n.m+n.
2019 AIME IIAIMEAIME II