MathDB

15

Part of 2007 AIME Problems

Problems(2)

An Equilateral Triangle

Source: AIME I 2007 #15

3/15/2007
Let ABCABC be an equilateral triangle, and let DD and FF be points on sides BCBC and ABAB, respectively, with FA=5FA=5 and CD=2CD=2. Point EE lies on side CACA such that ∠DEF=60∘\angle DEF = 60^\circ. The area of triangle DEFDEF is 14314\sqrt{3}. The two possible values of the length of side ABAB are p±qrp \pm q\sqrt{r}, where pp and qq are rational, and rr is an integer not divisible by the square of a prime. Find rr.
geometryratiotrigonometryquadraticsAMCAIMEUSA(J)MO
Tangent Circles

Source: AIME II 2007 #15

3/29/2007
Four circles ω,\omega, ωA,\omega_{A}, ωB,\omega_{B}, and ωC\omega_{C} with the same radius are drawn in the interior of triangle ABCABC such that ωA\omega_{A} is tangent to sides ABAB and ACAC, ωB\omega_{B} to BCBC and BABA, ωC\omega_{C} to CACA and CBCB, and ω\omega is externally tangent to ωA,\omega_{A}, ωB,\omega_{B}, and ωC\omega_{C}. If the sides of triangle ABCABC are 13,13, 14,14, and 15,15, the radius of ω\omega can be represented in the form mn\frac{m}{n}, where mm and nn are relatively prime positive integers. Find m+n.m+n.
geometryinradiusincentergeometric transformationhomothetycircumcircleratio