MathDB

7

Part of 2001 AIME Problems

Problems(2)

Triangle

Source:

12/6/2005
Triangle ABCABC has AB=21AB=21, AC=22AC=22, and BC=20BC=20. Points DD and EE are located on AB‾\overline{AB} and AC‾\overline{AC}, respectively, such that DE‾\overline{DE} is parallel to BC‾\overline{BC} and contains the center of the inscribed circle of triangle ABCABC. Then DE=m/nDE=m/n, where mm and nn are relatively prime positive integers. Find m+nm+n.
geometryincenterratioinradiusAMCAIMEnumber theory
Right Triangle and Circles

Source:

12/28/2006
Let △PQR\triangle{PQR} be a right triangle with PQ=90PQ=90, PR=120PR=120, and QR=150QR=150. Let C1C_{1} be the inscribed circle. Construct ST‾\overline{ST} with SS on PR‾\overline{PR} and TT on QR‾\overline{QR}, such that ST‾\overline{ST} is perpendicular to PR‾\overline{PR} and tangent to C1C_{1}. Construct UV‾\overline{UV} with UU on PQ‾\overline{PQ} and VV on QR‾\overline{QR} such that UV‾\overline{UV} is perpendicular to PQ‾\overline{PQ} and tangent to C1C_{1}. Let C2C_{2} be the inscribed circle of △RST\triangle{RST} and C3C_{3} the inscribed circle of △QUV\triangle{QUV}. The distance between the centers of C2C_{2} and C3C_{3} can be written as 10n\sqrt{10n}. What is nn?
geometryinradiusincenteranalytic geometrysimilar triangles